Invariant integrals in continuum mechanics

Presented at the International Conference on Contemporary Mathematical Problems in Mechanics and Their Applications (Moscow, November 11–15, 1987).     

Phase Equilibria and Crystal Structure of Complex Oxides in the La–Sr–Co–Ni– System

Phase equilibria in the La–Sr–Co–Ni–O system were studied in air at 1100°. The samples for the study were synthesized by the standard ceramic and citrate processes. The limiting solubility and structure of La_1-xSr_xCo_1-yNi_yO_3- and (La_1-xSr_x)₂Co_1-yNi_yO₄ solid solutions were determined by Xray powder diffraction analysis. La_1-xSr_xCo_1-yNi_yO₃- solid solutions with 0 x 0.5 have a distorted rhombohedral perovskitelike structure (R c space group). An increase in the strontium concentration reduces the rhombohedral distortions, and the compounds with x < 0.5 have an ideal cubic structure (Pm3m space group). (La_1-xSr_x)₂Co_1-yNi_yO₄ crystals have a tetragonal K₂NiF₄ type unit cell (I4/mmm space group). The relationships between unit cell parameters and compositions were obtained for singlephase La_1-xSr_xCo_1-yNi_yO_3- and (La_1-xSr_x)₂Co_1-yNi_yO₄ samples. The existence regions of La_1-xSr_xCo_1-yNi_yO₃- and La_1-xSr_x)₂Co_1-yNi_yO₄ solid solutions were distinguished on P–T phase diagrams.     

3-Lithiomethyl-4-phenylsydnone: A new type of organometallic derivatives of sydnones

An α-lithium derivative of 3-alkyl-substituted sydnone was first synthesized by direct metallation of 3-methyl-4-phenylsydnone withn-butyllithium. The reactivity of the compound obtained was studied. Reactions of 3-lithiomethyl-4-phenylsydnone with various electrophiles can serve as a convenient method for preparation of functionalized sydnones.     

Quantum model of the magnetic dynamics of single-domain particles for describing their magnetization curves and Mössbauer spectra in a weak magnetic field

A universal approach to describing the equilibrium magnetization curves and relaxation Mössbauer spectra of magnetic nanoparticles is proposed for consistent analysis of magnetometry and gamma-resonance experimental data, based solving a quantum-mechanical problem for a particle with spin S that has intrinsic magnetic anisotropy and is positioned in an external magnetic field.     

Equistrong heavy beam: Solving the problem of Galileo Galilei

In his main book “Discorsie Dimostrazioni Matematiche, Intorno a Due Nuove Scienze” published in 1638 by Elsevier in Leiden, Galileo Galilei, “the Father” of modern science, put the material science and strength of materials on the first place. He introduced the notions of stress and strength that have been fundamental since then. Moreover, in unison with Plato’s theory of forms he found out the perfect shape of a force-bent beam we call today equistrong. This discovery laid the foundation for search of other perfect elastic bodies as a continuation of Galilei’s work. There are no theorems of existence for equistrong bodies so that the quest for them is like a gold-digging. In what follows, the shapes of the following heavy, equistrong beams were found out: a) beam of constant thickness and of variable width, simply supported at both ends, b) beam clamped at one end and loaded at the other end while having either constant thickness and variable width, or constant width and variable thickness, and c) equistrong shape of the profile of aircraft wings accounting for gravity and lift loads. The shape of equistrong rod at buckling under a compressive force is found in the Euler’s problem. Equistrong structures possess minimum weight for given safety factor or maximum safety factor for given weight.