Fundamental Theory and Implementation of the Wang-O'Gara-Tucker Model for the Modeling of Fiber Orientation in Fiber Filled Injection Molded Thermoplastics

Abstract

The purpose of this master project is to implement the Wang-O'Gara-Tucker model to simulate the fiber orientation in fiber filled injection molded thermoplastics. A part of the extensive fundamental theory behind this model, and the implementations are treated in Part I of the thesis, the theory part. This part consists of the first eight chapters. In Chapter 1 an introduction to the thesis is given. Certain preliminaries to understand the scope and purpose of the project are treated. These preliminaries include the process of injection molding, the use of fiber filled thermoplastics, fiber orientation, past research, available commerical software, the interest of DSM in the subject of fiber orientation, the research method and approach and the contribution of this thesis. In Chapter 2 the notation and conventions used throughout this report are treated. As is a selection of the mathematical preliminaries. Chapter 3 starts off with the theory behind Jeffery's equation, a deterministic model for the fiber orientation of a single ellipsoidal particle in a fully viscous/Stokes flow. This model is extended by Folgar and Tucker with the addition of a fiber-fiber interaction term, using a probability density function. In Chapter 4 an approach using only partial information of the computationally inefficient Folgar-Tucker model is introduced. Chapter 5 treats the model that is the focus of this thesis, the Wang-O'Gara-Tucker model. It slows down the kinetics of the Folgar-Tucker model to adapt the theoretical results to the experimental results. In Chapter 6 the closure problem of the Folgar-Tucker tensor form is explained. Chapter 7 thoroughly treats the Fokker- Planck equation that stems from the Folgar-Tucker probability density form. A numerical scheme is derived to obtain a solution. In Chapter 8 a numerical scheme for solving the tensor form of the Folgar-Tucker model is derived. Should one not be interested in mathematical details, then proofs can be skipped easily. The idea of the models can be understood even without such details. If one is merely interested in the results of the implementation of the model, then Part I can be skipped entirely. The results are contained in Part II. Part II uses the implementations of Chapters 7 and 8 to model the fiber orientation for a simple shear flow field in Chapter 9 and for a uniaxial elonational flow in Chapter 10. Results of the probability density are interpreted for different parameter sets. The approximations of the tensor form are compared to the solutions of the probability density, using several closure relations. Part III concludes the theory and test models with conclusions and recommendations for future research. Part IV consists of some 'classical' proofs, instructions for the codes that were used to obtain the results in the thesis, the bibliography and the nomenclature. The classical proofs were put in the appendix to make sure that only information specific to the modeling of fiber orientation with the Wang-O'Gara-Tucker model is included in the main parts. With the addition of the instruction of the implementations to the thesis, it is intended to make the thesis offer an integrated approach to the Wang-O'Gara-Tucker model: the theory is presented, two test cases as examples are considered and the usage of the implementations is explained.