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 $\zeta ={\nu }_{\mathrm{ph}}$, if $\mu ⩽{\nu }_{\mathrm{ph}}$, where ${\nu }_{\mathrm{ph}}=3∕4$ is the size exponent for crumpled phantom membrane, or both exhibit an intrinsically anomalous roughness characterized by the universal local roughness exponent $\zeta =0.72\pm 0.04$ and the material dependent global roughness exponent $\alpha =\mu$, when $\mu >{\nu }_{\mathrm{ph}}$. The physical implications of these findings are discussed.

The bending of self-similar beams applying the Euler–Bernoulli principle is studied in this paper. A generalization of the standard Euler–Bernoulli beam equation in the ${\mathcal{F}}_{d_{\mathcal{H}}}^3$ continuum using local fractional differential operators is obtained. The mapping of a bending problem for a self-similar beam into the corresponding problem for a fractal continuum is defined. Displacements, rotations, bending moments and shear forces as functions of fractal parameters of the beam are estimated, allowing the mechanical response for self-similar beams to be established. An example of the structural behavior of a cantilever beam with a load at the free end is considered to study the influence of fractality on the mechanical properties of beams.

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\ifmmode\pm\else\textpm\fi{}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.

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 $(D_l=2.64\pm 0.05)$ 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 $D<D_l$. So intrinsically anomalous self-similarity is expected to be an essential feature of randomly folded thin matter.

We study the dynamics of the seismic activity in Mexico within a framework of dynamic scaling approach to time series fluctuations, recently suggested by Balankin (Phys. Rev. E, 76 (2007) 056120). We found that the relative seismic activity and the long-sampled fluctuations of seismic activity both display a self-affine invariance within a wide range of consecutive seismic evens. Furthermore, we found that the long-sampled fluctuations of seismic activity obey the dynamic scaling ansatz analogous to the Family-Vicsek dynamic scaling ansatz in the theory of kinetic roughening of moving interfaces. These findings imply that the records of recurrent seismic events possess hidden, long-term correlations associated with the scaling dynamics of seismic activity fluctuations.