Andrei L. Smirnov

Effect of initial conditions on the dispersion dynamics of a diffusing substance

This work continues the studies on the diffusion of a substance over a water surface, in particular, the effect of nonuniformity in the initial distribution of a substance on the dynamic characteristics of a pollution spot has been investigated. A pollution spot is understood to mean a water surface area in which the concentration of a diffusing substance is higher than a specified threshold value. The analytical solutions of boundary-value problems have been found by the Fourier method in special functions for the equation of diffusion in unlimited areas. Asymptotic and numerical methods are used for their analysis. It has been concluded that the initial distribution of a polluting substance over the surface has a slight effect not only on the lifetime of a pollution spot but also on its maximum radius at the same volume of pollution. The maximum size of a pollution spot and the time moment at which this size is attained have been found in the case of a uniform substance distribution.

Free vibrations of perforated thin plates

The report includes the updated results of analysis of buckling and vibrations of thin-walled structures previously reported in [1, 2]. Free vibrations of a thin plate with multiple cut-outs are modeled by means of analytic and numerical methods. The effect of the total area of holes, their positions and shapes, the shape of a plate shape and the plate side ratio on natural frequencies and modes of free vibrations is studied. It appeared that the natural frequencies may either increase or decrease with the total cut-outs area since the holes affects both the stiffness and the mass of a plate. The interaction between holes and modes of vibrations plays the crucial role. For small cutout area the asymptotic formula for natural frequencies has been obtained and the asymptotic results have been compared with the results of numerical analysis.

Asymptotic methods in mechanics of solids

Asymptotic Estimates.- Asymptotic Estimates for Integrals.- Regular Perturbation of ODEs.- Singularly Perturbed Linear ODEs.- Linear ODEs with Turning Points.- Asymptotic Integration of Nonlinear ODEs.- Bibliography.- Index.

The three-dimensional problem of the axisymmetric deformation of an orthotropic spherical layer

A 3D problem of the deformation of an elastic orthotropic spherical layer that is subjected to normal pressure applied to its outer and inner surfaces is analyzed. Asymptotic first-order approximation solutions are obtained for a slightly orthotropic layer for which the elastic moduli in the meridional and circumferential directions have similar values. The solutions that are obtained are used for analyzing the scleral shell under intraocular pressure; however, they can also be used for solving the inverse problem of analyzing the stress–strain state of a human eye during intravitreal injections. The influence that the meridional and circumferential elastic moduli have on the magnitudes of changes in the relative layer thickness and in the length of the anteroposterior eye axis due to elevated intraocular pressure is studied.

ASYMPTOTIC ANALYSIS OF DEFORMATIONS OF THE SLIGHTLY ORTHOTROPIC SPHERICAL LAYER UNDER NORMAL PRESSURE

The deformation of the orthotropic spherical layer under normal pressure applied on the outer and inner surfaces is analyzed. The layer is assumed to be slightly orthotropic, it permits to apply asymptotic methods. The equations of zeroth and first approximations are derived. For the shell, which is much softer in the transverse direction than in the tangential plane, one gets singularly perturbed boundary value problem. Solving this problem in the zeroth approximation the asymptotic formula for the change of the relative layer thickness under normal pressure is obtained. Also the effect of Poisson ratio and the layer thickness on the deformation is studied. For the cases of the thick and thin layers the last formula may be simplified. The asymptotic results well agree with the exact solution. The developed formulas are used in analysis of the scleral shell under intraocular pressure and may also be used in solution of the inverse problem, i.e. in analysis of the stress-strain state of a human eye under injection. The solution of the problem helps to estimate the mechanical parameters of the sclera, i.e. to find the ratio of the tangential and transversal Young moduli using clinical data for the sclera thickness change.

Asymptotic Integration of Nonlinear Differential Equations

There are several types of asymptotic expansions for the solutions of nonlinear differential equations. Regularly perturbed nonlinear equations were considered in Chap. 3.

Asymptotic Estimates for Integrals

Mechanical problems can be described by differential equations, the solutions of which often cannot be expressed by elementary functions, but have an integral representation.

Singularly Perturbed Linear Ordinary Differential Equations

In this chapter, we study systems of linear differential equations with variable coefficients.

Singularly Perturbed Linear Ordinary Differential Equations with Turning Points

In this chapter, we consider systems of linear ordinary differential equations with variable coefficients and a small parameter \(\mu \) in the derivative terms.

Regular Perturbation of Ordinary Differential Equations

In this chapter we find asymptotic solutions of regularly perturbed equations and systems of equations, to which problems in mechanics are reduced. We consider Cauchy problems, problems for periodic solutions and boundary value problems.