We study the effect of folding ridges on the scaling properties of randomly crumpled sheets of different kinds of paper in the folded and unfolded states. We found that the mean ridge length scales with the sheet size with the scaling exponent mu determined by the competition between bending and stretching deformations in the folded sheet. This scaling determines the mass fractal dimension of randomly folded balls D{M}=2/mu. We also found that surfaces of crumpled balls, as well as unfolded sheets, both display self-affine invariance with zeta=nu{ph}, if mu < or =nu{ph} , where nu{ph}=34 is the size exponent for crumpled phantom membrane, or both exhibit an intrinsically anomalous roughness characterized by the universal local roughness exponent zeta=0.72+/-0.04 and the material dependent global roughness exponent alpha=mu, when mu>nu{ph}. The physical implications of these findings are discussed.

We found that randomly folded thin sheets exhibit unconventional scale invariance, which we termed as an intrinsically anomalous self-similarity, because the self-similarity of the folded configurations and of the set of folded sheets are characterized by different fractal dimensions. Besides, we found that self-avoidance does not affect the scaling properties of folded patterns, because the self-intersections of sheets with finite bending rigidity are restricted by the finite size of crumpling creases, rather than by the condition of self-avoidance. Accordingly, the local fractal dimension of folding structures is found to be universal (Dl=2.64+/-0.05) and close to expected for a randomly folded phantom sheet with finite bending rigidity. At the same time, self-avoidance is found to play an important role in the scaling properties of the set of randomly folded sheets of different sizes, characterized by the material-dependent global fractal dimension D

We study the statistical topology of folding configurations of hand folded paper balls. Specifically, we are studying the distribution of two sides of the sheet along the ball surface and the distribution of sheet fragments when the ball is cut in half. We found that patterns obtained by mapping of ball surface into unfolded flat sheet exhibit the fractal properties characterized by two fractal dimensions which are independent on the sheet size and the ball diameter. The mosaic patterns obtained by sheet reconstruction from fragments of two parts (painted in two different colors) of the ball cut in half also possess a fractal scale invariance characterized by the box fractal dimension DBF=1.68 ± 0.04 , which is independent on the sheet size. Furthermore, we noted that DBF, at least numerically, coincide with the universal fractal dimension of the intersection of hand folded paper ball with a plane. Some other fractal properties of folding configurations are recognized.

No abstract

We study the lateral deformations of randomly folded elasto-plastic and predominantly plastic thin sheets under the uniaxial and radial compressions. We found that the lateral deformations of cylinders folded from elasto-plastic sheets of paper obey a power law behavior with the universal Poisson's index 01 . 0 17 . 0 ± = ν , which does not depend neither the paper kind and sheet sizes (thickness, edge length), nor the folding confinement ratio. In contrast to this, the lateral deformations of randomly folded predominantly plastic aluminum foils display the linear dependence on the axial compression with the universal Poisson's ratio 01 . 0 33 . 0 ± = e ν. This difference is consistent with the difference in fractal topology of randomly folded elasto-plastic and predominantly plastic sheets, which is found to belong to different universality classes.The general form of constitutive stress-deformation relations for randomly folded elastoplastic sheets is suggested.