Fractal balls

Abstract

To the best of our knowledge, the analysis of densely folded media has not deserved special attention. The stress and strain analysis of this type of structures involves considerable difficulties concerning very strong non-linear effects. This paper presents a theory that could be classified as a geometric theory of folded media, in the sense that it ultimately leads to a kind of geometric constitutive law, or, in other words, a law that establishes the relationship between the geometry of the folded media and other variables such as the confinement capacity and the plastic strain energy. The discussion presented here is restricted to the particular case of compact balls produced by crushing together very thin plates or sheets. It is shown that both the geometry of the folded sheet and the plastic work density can be used as self-similarity tests. These criteria are equivalent for the case of thin plates or sheets made of the same material and with the same thickness. For the general case, the geometry of the folded sheet is not valid anymore as similarity criterion but there are strong arguments in favor of the plastic work density as a general criterion. If self-similarity is obtained for a ball set resulting from crumpling thin plates or sheets, it is possible to define two variables, the packing capacity and the slenderness ratio, that are related according to a power law. That is, the balls have a fractal representation. The power law scaling is derived from the mass conservation principle. The theory is claimed to be valid provided that certain assumptions referring to the geometry and material properties are satisfied. The results have shown that the theory is coherent and worthwhile of experimental validation. Some applications are suggested. A possible challenging investigation is related to the optimal geometry of biological membranes.