# Mechanics

## Cylindrical shell with a circular hole under various loads: Comparison of analytical and numerical solutions

In this paper, the authors present the results of calculations of the stress field of a cylindrical shell weakened by a circular hole and under the influence of various loads: uniaxial tension, internal pressure and torsion. Six simplified equations of the theory of cylindrical shells with a high variability index (coinciding with the equations of the theory of shallow shells) are reduced to an equation of mathematical physics with respect to the stress potential, which is solved by the Fourier method.

## The new algorithm of quasi-optimal reorientation of a spacecraft

The classical problem of optimal control of the attitude maneuver of a spacecraft as a rigid body of arbitrary dynamic configuration under arbitrary boundary conditions for the angular position and angular velocity of a spacecraft without restriction on the control vector function and with a fixed transition time is considered. As a criterion of optimality, the functional of the energy spent on the rotation of a spacecraft is used.

## Influence of a polymeric infiltrant on the density of enamel white spot lesions

In modern dental practice, treatment of early stages of caries is possible using minimally invasive intervention. In this work, using X-ray computed microtomography (micro-CT), an *ex vivo* non-destructive study of the density of white spot lesions was carried out before and after the application of a polymer infiltrant.

## On the physical equations of a deformable body at the loading step with implementation based on a mixed FEM

To obtain the deformation matrix of the prismatic finite element at the loading step, taking into account the physical nonlinearity, three variants of physical equations were used. In the first variant, the defining equations of the theory of plastic flow are implemented, according to which the increment of deformations is divided into elastic and plastic parts. The increment of elastic deformations is related to the increments of stresses by Hooke's law.

## Elastic-plastic deformation of nanoplates. The method of variational iterations (extended Kantorovich method)

In this paper, a mathematical model is constructed based on the deformation theory of plasticity for studying the stress-strain state of Kirchhoff nanoplates (nanoeffects are taken into account according to the modified moment theory of elasticity). An economical and correct iterative method for calculating the stress-strain state of nanoplates has been developed — the method of variational iterations (the extended Kantorovich method).

## Contact problem for functionally graded orthotropic strip

Within the framework of plane elasticity, the equilibrium problem for an inhomogeneous orthotropic elastic strip under the action of a stamp with a smooth base is investigated. Based on the Fourier transform, a canonical system of differential equations with variable coefficients with respect to transformants of the displacement vector and stress tensor components is constructed.

## Dynamic thermal stability of a geometrically irregular shallow shell of constant torsion under the action of a load periodic by its time coordinate

In the framework of a Love type model, a geometrically irregular isotropic shallow constant torsion shell is considered. It is based on a strict continuum-shell-rib model. It is assumed that the geometrically irregular shell is heated to a constant temperature $\theta_0$, two opposite edges are exposed to a tangential load periodic by its time coordinate, the amplitude and frequency of which are known ($p(t)=p_0 \cos \vartheta t$).

## Bending of an elastic circular three-layer plate in a neutron flux by a local load

The bending of an elastic circular three-layer plate asymmetric in thickness by local loads uniformly distributed in a circle in a neutron current is considered. Polyline hypotheses are used to describe the kinematics of the package. Kirchhoff's hypotheses are valid in thin load-bearing layers. In a relatively thick incompressible filler, Timoshenko's hypothesis about the straightness and incompressibility of the deformed normal is fulfilled. The work of the tangential stresses of the filler is taken into account. Deformations are small.

## Evaluation of the influence of white spot lesion on the mechanical properties of human tooth enamel and dentine

In the present paper the influence of early caries (white spot lesion) on the mechanical properties of human tooth enamel and dentine was *ex vivo* investigated. Optical microscopy made it possible to study the shape of the enamel caries area on a prepared longitudinal section of a human molar.