Correlation of thermal conductivities of unidirectional fibrous composites using local fractal techniques

The arrangement of fibers strongly influences heat conduction in a composite. Traditional approaches using unit cells to describe the fiber arrangements work well in the case of ordered arrays, but are not useful in the context of disordered arrays, which have been analyzed in the literature by statistical means. This work presents a unified treatment using the tool of local fractal dimensions (although, strictly speaking, a composite cross section may not be an exact fractal) to reduce the geometric complexity of the relative fiber arrangement in the composite. The local fractal dimensions of a fibrous composite cross section are the fractal dimensions that it exhibits over a certain small range of length scales. A generalized unit cell is constructed based on the fiber volume fraction and local fractal dimensions along directions parallel and transverse to the heat flow direction. The thermal model resulting from a simplified analysis of this unit cell is shown to be very effective in predicting the conductivities of composites with both ordered as well as disordered arrangement of fibers. For the case of square packing arrays, the theoretical result of the present analysis is identical to that of Springer and Tsai.