Z. Yong et Mt. Hanson, A RATIONAL SOURCE OF PLANE FRACTALS AND ITS APPLICATION TO FRAGMENTATION ANALYSIS OF THIN PLATES, Chaos, solitons and fractals, 7(1), 1996, pp. 31-40
It is demonstrated in this paper that the Euler equation which corresp
onds to the subdivision of a torus into a polyhedron is a rational sou
rce of plane fractals. The vertex orders and polygon edges within a pl
ane fractal are an invariant of the fractal. The process of forming a
fractal is intimately related to the topological and infinitely divisi
ble properties of the solutions for the Euler equation. Based on this
equation it is shown that only four linear plane fractals are availabl
e and the boundary of a plane fractal is a fractal curve, such as the
Koch curve. Certain relations between a plane fractal and a fractal fo
rmed by the Julia set are also discussed with the aid of a new plane f
ractal pattern given in this work. As an application of the present an
alysis, brittle fragmentation of a thin plate is considered. Present r
esults provide a new expression for estimating the average size of a f
ragment based on an energy balance principle and an idea about quasi-i
dentical plane fractals. Present theoretical analysis is in agreement
with experimental results and previous investigations.