Numerical prediction of three-dimensional fiber orientation during mold filling is based on an equation of motion for rigid particles in a fluid suspension.
The hydrodynamic influence on particle motion is described by Jeffery's equation assuming infinite aspect ratio. This theory strictly applies to dilute suspensions but has been shown to provide useful qualitative agreement with experimental data.
The interaction term has been proposed by Folgar and Tucker and is incorporated to model the randomizing effect of mechanical interactions between fibers. It has the form of a diffusion term with the frequency of interaction being proportional to the magnitude of the strain rate. The effect of the interaction term is to reduce highly aligned orientation states predicted by Jeffery's model for some flow conditions, providing improved agreement with experimental observations.
Calculation of three dimensional fiber orientation is performed concurrent with the mold filling analysis on the same finite element mesh. Each triangular element may be considered as consisting of several layers subdividing the local molding thickness. Each layer is identified by the grid point through which it passes. The midplane of the molding passes through grid point 1. An orientation solution is calculated at each layer for each element in the mesh. In this way it is possible to observe the variation in orientation distribution on a set of planes parallel to the mold surface through the cross-section of the molding.
The three-dimensional orientation solution for each element is described by a second order tensor. For graphical representation, the eigenvalues and eigenvectors of the orientation tensor are generated. The eigenvectors indicate the principal directions of fiber alignment and the eigenvalues give the statistical proportions (0 to 1) of fibers aligned with respect to those directions. This information is used to define an orientation ellipsoid which fully describes the alignment distribution of fibers for each element. A general orientation ellipsoid is shown in the figure below.
For display purposes, this 3D ellipsoid is projected onto the plane of each element to produce a plane ellipse. This creates a useful representation of orientation distribution, since the gapwise orientation components eliminated by projection are usually small. In this representation a near random distribution is displayed as an ellipse tending to a circle while, for a highly aligned distribution, the ellipse degenerates to a line.
Jeffery first modeled the motion of a single fiber immersed in a large body of incompressible Newtonian fluid. Jeffery's model applies only to suspensions that are so dilute that any inter-fiber interactions (even hydrodynamic interactions) are negligible.
Considering fibers of diameter (d) and length (L), with an aspect ratio (L/d), a fiber concentration by volume (c) (or volume fraction ) and having a uniform length distribution, a typical concentration classification scale is:
For semi-concentrated suspensions, a model has been proposed by Dinh and Armstrong. The orientation of the fiber follows the bulk deformation of the fluid with the exception that the particle cannot stretch.
Adding the rotary diffusion term to account for the fiber interactions has been found to improve the orientation predictions, since Jeffery's equation alone does not give qualitatively accurate predictions for fiber orientation.
The derivative for a fourth order tensor contains a sixth order orientation tensor and so on. The only way to develop a suitable derivative is to approximate the fourth order tensor in terms of a second order tensor.
This approximation is called a closure approximation. Various approximations have been tested by Advani and Tucker. However, the presence of the approximation itself may introduce some error into the simulation results. So the closure approximation is the most challenging problem associated with this model. No value of can make the fiber orientation model expression fit all the orientation model components.