CIRCULAR CHAOS GAME REPRESENTATION OF 1-D CHAOS AND ITS RELATION TO THE COMPLEX WEIERSTRASS FUNCTION

Barnsley's chaos game which is originally played on a triangular field is generalized to a circular field and is used to visualize one-dimensional (1-D) fully developed chaos. It is shown that when the 1-D map used is the shift map or its extension (the r-adic map) the game point is exactly represented by the complex Weierstrass function.

Disk and Strip Forging with Side Surface Foldover—Part 2: Evaluation of the Upper-Bound Solutions

The upper-bound solutions developed in Part 1 are evaluated with regard to their ability to produce a lower value for required power (load, pressure, or work). Comparisons made with existing solutions such as the triangular field solution and one-zone bulge solution show that for strip, each solution has a domain of geometry and friction in which it is superior. The new solution produces a lower upper-bound for conditions of high interface friction and relatively thin specimen, the area where foldover is the observed mode of flow. For solid cylindrical disks, the solution fails to improve upon existing analyses, but comes sufficiently close to warrant additional study. After evaluation, these solutions were then used in an incremental technique to model the geometry and flow as a function of reduction in height. Results appear most encouraging, and the relative simplicity of the technique when compared with present alternatives is quite attractive.