An anisotropic rotary diffusion model for fiber orientation in short- and long-fiber thermoplastics

The Folgar–Tucker model, which is widely-used to predict fiber orientation in injection-molded composites, accounts for fiber–fiber interactions using isotropic rotary diffusion. However, this model does not match all aspects of experimental fiber orientation data, especially for composites with long discontinuous fibers. This paper develops a fiber orientation model that incorporates anisotropic rotary diffusion. From kinetic theory we derive the evolution equation for the second-order orientation tensor, correcting some errors in earlier treatments. The diffusivity is assumed to depend on a second-order space tensor, which is taken to be a function of the orientation state and the rate of deformation. Model parameters are selected by matching the experimental steady-state orientation in simple shear flow, and by requiring stable steady states and physically realizable solutions. Also, concentrated fiber suspensions align more slowly with respect to strain than models based on Jeffery's equation, and we incorporate this behavior in an objective way. The final model is suitable for use in mold filling and other flow simulations, and it gives improved predictions of fiber orientation for injection molded long-fiber composites.