Polymer Blends and Composites in Multiphase Systems - American

17 Prediction and Control of Fiber Orientation in Molded Parts WILLIAM C. JACKSON, FRANCISCO FOLGAR, and CHARLES L. TUCKER III

Downloaded by CORNELL UNIV on June 1, 2017 | http://pubs.acs.org Publication Date: May 1, 1984 | doi: 10.1021/ba-1984-0206.ch017

Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 The problem of predicting the fiber orientation pattern in a part molded from a short-fiber reinforced polymer is formulated. The solution requires that a governing equation for the fiber orientation distribution function be integrated along the paths of fluid particles during mold filling. A set of simplifying assumptions is presented for thin compression-molded parts with fibers much longer than the part thickness. Example calculations agree with familiar qualitative rules for fiber orientation and with quantitative experiments on sheet molding compounds.

POLYMERS WITH DS ICONTN I UOUS FIBER REINFORCEMENT

are attractive mate rials because they combine stiffness, strength, and light weight. I n addi tion, they can be manufactured by l o w labor content processes such as injection molding, compression molding, and extrusion. Virtually all pro cessing techniques for these materials can orient the fibers preferentially, a feature that controls many physical properties of the resulting composite. Stiffness, strength, thermal expansion, and thermal conductivity are among the many properties of a composite that change with fiber orienta tion. In many cases the changes are major. F o r example, the elastic modu lus of a typical glass fiber-polyester matrix composite more than doubles i n one direction and drops by a factor of four i n the perpendicular direction as the fiber orientation changes from planar random to fully aligned ( i ) . Mechanical property variations of this order have significant impact on the performance of the materials, but this fact has largely been ignored by designers and processors. I n many instances, fiber orientation has been treated as an undesirable source of mechanical property variation, and processors have sought to minimize any preferential orientation. A much more fruitful approach would be to control the fiber orientation to improve material performance. W h e n this approach can be taken by judicious de0065-2393/84/0206-0279$06.25/0 © 1984 American Chemical Society

Han; Polymer Blends and Composites in Multiphase Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1984.


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POLYMER BLENDS AND COMPOSITES IN MULTIPHASE SYSTEMS

sign of the part and careful choice of processing conditions, these improve ments can be had w i t h no increased costs. E v e n the ability to predict how fibers w i l l be oriented i n a part would be useful, because, otherwise, one must make a mold, manufacture prototype parts, and test them to be sure that they w i l l work. In fact, we currently have the ability to predict mechanical properties of short-fiber composites once the fiber orientation distribution is known. Elastic constants (1-5), thermal expansion coefficients (1-3), and tensile strength (5, 6) can be accurately predicted. Even the nonlinear stressstrain curve of a brittle matrix composite can be predicted (7). Other prop erties such as notch sensitivity and viscoelastic response have been shown experimentally to depend on fiber orientation (I, 8). Even a simple quanti tative approach to account for the orientation distribution would improve design. In this chapter, we consider the problem of predicting fiber orienta tion distributions as a function of processing variables for molded parts. W e first present a general approach to the problem and then discuss its application to compression molding. The goal of the calculation is to pre dict the entire fiber orientation distribution at every point i n the finished part. The suggestion has been made that one should try to predict or corre late two particular weighted integrals of the orientation distribution as a function of processing conditions, because, under certain symmetry condi tions, one can predict stiffness constants and thermal expansion coefficients by knowing only the values of the two integrals (I). However, not all prop erties can be predicted from this information. W e have chosen the more general approach of calculating the entire orientation distribution func tion.

Background The motion of a single rigid ellipsoidal particle i n a deforming viscous fluid was originally derived by Jeffrey (9). His equations have been shown (10) to apply to rigid particles w i t h other shapes as well (e.g., cylinders). A l though the particle motion is affected somewhat by viscoelasticity i n the fluid (II), these equations can be applied with some confidence to polymer processing problems where few fibers are present. Certainly the high vis cosity of polymer melts and resins and the small diameter of reinforcing fibers validate the assumption of negligible inertia. Consider the simplified case of a rigid cylindrical particle whose axis lies i n the x-y plane and a flow field for w h i c h the unperturbed flow is planar i n the x-y plane. The fiber orientation is described by the single angle φ, as shown i n Figure 1. Jeffrey's equations give the motion of this particle as


17.

JACKSON ET AL.

Prediction and Control of Fiber Orientation

281

φ = [t*l(fZ + 1)] { - sin φ cos φ (dvjdx) - sin φ (dvjdy) 2

+ cos φ (dVy/dx) + sin φ cos φ (dv ldy)} 2

y

- [II(ίξ + 1)] { - sin φ cos φ(3ν Ι3χ) + cos φ (dvjdy) 2

χ

- s i n φ(ον Ι3χ) + sin φ cos φ (dv /dy)} 2

ν

(1)

y

where φ is the angular velocity of the particle, and v and v are the fluid velocity components. The quantity r is the equivalent ellipsoidal aspect ratio (10) and is on the order of the length-to-diameter ratio of the cylinder. This equation shows that fiber motions are controlled by the kine matics of the flow. In simple and planar extensional flows, the fiber rotates toward the direction of stretching; when the fiber is aligned i n that direc tion it is in stable equilibrium. In simple shear flow, the fiber rotates w i t h a periodic motion, but spends much more time oriented i n the direction of the flow than i n other directions. Hence, we have the familiar rules of thumb: (1) shear flows orient fibers i n the direction of flow, and (2) exten sional flows orient fibers i n the direction of extension. These equations of fiber motion have been used by Harris and Pittman (12) to analyze the fiber orientation of a dilute suspension of fibers i n a converging nozzle. They solved for the velocity distribution i n the nozzle, then used the deformation field from the solution to calculate fiber orienta tion changes. Their calculations were i n excellent agreement w i t h experiments. Some calculations of fiber orientation i n processing were also perx

y


e

y

Figure 1.

Definition of the angle φ that describes the orientation of a fiber lying in the x-y plane.


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POLYMER RLENDS AND COMPOSITES IN MULTIPHASE SYSTEMS

formed by Modlen (13), for fibers i n a plastically deforming matrix, and by Nicolais et al. (14), for postextrusion drawing of a glass fiber/polystyrene composite. Both works involved primarily extensional strains of dilute sus pensions, and both used a simple theory of fiber motion that assumes a fiber rotates like a fluid line. Their simple theory is i n fact identical to Equation 1 as r -» oo. This theory does not predict the observed periodic motion of fibers i n simple shear flow, but is very close to the real motion of fibers i n extensional flows. Hence, the agreement between theory and experiment i n these two works is not surprising. These works provide the pattern for predicting fiber orientation dur ing processing: first find the deformation kinematics, then apply the laws of fiber motion. There are two problems w i t h applying this pattern to molding of composite materials. One is that the well-known equations for the motion of a single fiber cease to apply once the concentration of fibers exceeds a very low level. A t concentrations on the order of (1/f/) and above, ry being the ratio of fiber length to diameter, fibers interact w i t h one another, and each interaction causes a rapid change i n fiber orientation (15). Although some information about interactions is available (16), no theory of the mechanics of interactions capable of solving fiber orientation problems is yet available. The second problem is solving for the deformations accumulated by molding a part into a complex shape. In the most general case, the solution of the mold filling pattern is coupled with the fiber orientation problem, because the rheological properties of a fiber suspension are a function of fiber orientation. For example, sheet molding compound, w h i c h has nearly a planar random orientation distribution, has anisotropic viscosity. The ex tensional viscosity i n the plane of the sheet can be 100 times greater than the shear viscosity across the thickness of the sheet (17). A detailed rheologi cal model describing this phenomenon has been formulated by D i n h and Armstrong (18). Their model provides insight into the rheology of concen trated fiber suspensions. It does use a fiber orientation l a w that is i n conflict with our recent experimental data. The solution of fluid mechanics prob lems using this type of model is very difficult. Various experimental observations of fiber orientation i n processing situations have been made (19-28). Notably, no conflicts occur between experimentally observed fiber orientations and qualitative predictions based on the ideas discussed so far. Where the overall flow is converging, such as at the entrance to an extrusion die or an injection mold gate, fibers align i n the direction of flow in response to the stretching motion of the flow. Where the flow is diverging, such as where the flow enters an injec tion or transfer mold, fibers align transverse to the flow, i n the direction of stretching. For simple shearing flows, such as flow i n capillary tubes and i n parts of mold cavities, fibers align i n the direction of flow i n response to the shearing deformation.


e

2



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Prediction and Control of Fiber Orientation

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Fiber orientation was reported to have some dependence on the flow rate (and hence the strain rate). For instance, Crowson et al. (29) report that the degree of fiber orientation i n a capillary tube increases as the flow rate increases. Changes i n fiber orientation with filling rate i n thermoplas tic injection molding have also been reported by Bright et al. (25). A t first this evidence might seem i n conflict with the previous observation that changes i n fiber orientation should depend on the strain accumulated by the material but not on the strain rate at which it occurs. However, these workers only observe a rate dependence of fiber orientation under condi tions where changes i n the flow rate also change the kinematics of the flow and cause different parts of the material to undergo different strains. F o r instance, i n their capillary die equipment, Crowson et al. (29) report that the fiber orientation i n the capillary changes as the flow moves from the Newtonian region to the power-law region and changes the velocity profile and the distribution of strain rate across the tube. The extensional flow at the capillary entrance produces more highly aligned fibers than a steady state shearing flow. The shear flow i n the capillary tube actually disorients the fibers somewhat, and the flatter velocity profile at high flow rate causes less shear strain and has better orientation. For injection moldings, the ve locity profile changes because of rheological properties that change w i t h strain rate and temperature. As the flow rate is decreased, heat transfer becomes proportionally more important, and more orientation is "frozen i n " near the mold walls. The differences i n velocity profiles serve to explain the observed differences i n fiber orientation. Thus, no conflict exists be tween this type of experimental evidence and the idea of a strain-dependent orientation pattern. Although some rules for predicting orientation patterns based on the preceding ideas do exist (26), we need the ability to make quantitative pre dictions of fiber orientation; to predict the entire orientation distribution, not just the principal direction; and to do so on a firm fundamental basis. W e present a method for these predictions, beginning w i t h an appropriate model for fiber orientation behavior i n concentrated suspensions. The spe cific application of this model to the compression molding of thin parts is then shown as an example of how this type of calculation can be done. Theory Basic Fiber Orientation L a w . Experiments reveal that, i n a concen trated suspension of fibers, the motion of an individual fiber is not deter mined, as i n the dilute case (Equation 1), simply by its angle and by the flow field. Presumably the surrounding fibers also play a role through some type of interaction. In the absence of a detailed theory of interactions, we have chosen a probabilistic model that gives not the individual fiber orien tations but the probability density function of fiber orientation. The sus-


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POLYMER BLENDS AND COMPOSITES IN MULTIPHASE SYSTEMS

pension is viewed as a continuum, and the probability density function is taken to be a continuous function of space and time. For the case of a pla nar fiber orientation distribution, one can define the density function φ (φ) such that φ (φ) άφ is the probability that a fiber w i l l be oriented between the angles φ and φ + άφ. This definition is consistent with the usual definition of a probability density function; the probability of a fiber lying between angles φχ and φ (or the expected fraction of fibers oriented between φχ and φ ) is 2

2

Ρ(φ Downloaded by CORNELL UNIV on June 1, 2017 | http://pubs.acs.org Publication Date: May 1, 1984 | doi: 10.1021/ba-1984-0206.ch017

λ

Polymer Blends and Composites in Multiphase Systems - American

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