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.

Interaction of Alfven waves in the vicinity of multiple characteristic roots is considered. The asymptotic behavior for solutions of the Cauchy problem for the linearized equations of magnetohydrodyna

A new approach for solving the fractal Euler-Bernoulli beam equation is proposed. The mapping of fractal problems in non-differentiable fractals into the corresponding problems for the fractal continuum applying the fractal continuum calculus (${\mathcal{F}}_{d_{\mathcal{H}}}^3$-CC) is carried out. The fractal Euler-Bernoulli beam equation is derived as a generalization using ${\mathcal{F}}_{d_{\mathcal{H}}}^3$-CC under analogous assumptions as in the ordinary calculus and then it is solved analytically. To validate the spatial distribution of self-similar beam response, three different classical beams with several fractal parameters are analysed. Some mechanical implications are discussed.

The role of oxygen adsorption in the formation of varistor characteristics in ZnO:Mn ceramics was investigated. The samples were made of the mixture of ZnO powder with MnSO₄ aqueous solution, sintered in air for 3 h at 1100–1300 °C and furnace cooled. Mn content in the initial mixture was of 0.1 at.%. The density of prepared ceramics was of 5.37–5.41 g/cm³. XRD patterns indicated hexagonal wurtzite structure of obtained samples. The incorporation of manganese atoms in ZnO lattice as Mn_Zn²⁺ centres was proven by EPR spectroscopy. Current–voltage (CV) characteristics of as-prepared samples and of the same ones kept in N₂ flow, vacuum or oxygen atmosphere were compared. The increase of the sample conductivity and the decrease of CV characteristic nonlinearity after the exposure to nitrogen or vacuum were observed, whereas the action of oxygen resulted in the opposite effects. Obtained results testify that the adsorption of ambient oxygen contributes essentially to the formation of interface potential barriers in ZnO:Mn ceramics. The analysis of the available data has led to the conclusion that a thin layer of some Mn-related phase is formed at grain boundaries under sintering. This layer separates adjacent ZnO grains, which allows the diffusion of oxygen along grain boundaries and its adsorption on ZnO grain surface.

In this paper, we study the creep phenomena for self-similar models of viscoelastic materials and derive a generalization of the Kelvin–Voigt model in the framework of fractal continuum calculus. Creep compliance for the Kelvin–Voigt model is extended to fractal manifolds through local fractal-continuum differential operators. Generalized fractal creep compliance is obtained, taking into account the intrinsic time $\tau$ and the fractal dimension of time-scale $\beta$. The model obtained is validated with experimental data obtained for resin samples with the fractal structure of a Sierpinski carpet and experimental data on rock salt. Comparisons of the model predictions with the experimental data are presented as the curves of slow continuous deformations.