Eva B. Voronkova

Nonclassical Shell Theories in Ocular Biomechanics

The stress-strain state of a pressurized spherical shell and circular cylinder are studied by means of the exact 3D theory of elasticity and the 2D approximate shell theories of moderate thicknesses. Both the sphere and the cylinder are made of transversally isotropic material. The first problem models the changes in the fluid pressure inside the human eye due to injected additional volume of liquid. The pressurized cylinder is one of simplest model of the human vessels. The algebraic relationships for deflections and stresses are derived. Both described shell theories take into account the variation of the shell thickness, which can be important for soft materials, e.g. the human eye’s tissue. The asymptotical analysis of the exact 3D solutions has been performed and the accuracy of the approximate solutions, obtained with the approximate theories is analyzed. The effect of the thickness changes are also discussed.

Models of shells and plates in the problems of ophthalmology

This review is devoted to mathematical models created jointly with ophthalmologists. Models for calculating the stress-strain state of an eye shell after surgeries related to the treatment of retinal detachment and models of the theory of accommodation have been described briefly. Mathematical models describing the determination of the actual intraocular pressure (IOP) using application techniques have been discussed. Models making it possible to assess the effect of deviations of the shapes of the cornea and sclera from a spherical shape based on the IOP parameters and the effect of the cornea thickness on them have been also considered. It has been noted that models of ocular biomechanics helped in obtaining a number of new results in mechanics of solids, for example, in solving the problem on the stability of a spherical shell under a concentrated force and normal internal pressure, the stability of an axisymmetric equilibrium form of orthotropic nonuniform circular plates under normal pressure, the problem on the stability of a segment of an orthotropic shell under normal internal pressure and an applied load with a flat base, and solving problems of deformation of transversely isotropic spherical and cylindrical layers under internal and external pressures. The comparison of these solutions with those obtained using nonclassical shell theories made it possible to assess the precision of some theories.

On natural frequencies of transversely isotropic circular plates

The paper discusses the impact of the material properties of transversely isotropic circular plates on its natural frequencies. Two refined theories of plates have been used to analyze the free vibration behavior of homogeneous plates. Both theories take into account normal and rotary inertias. Fundamental frequencies for plates with radial inhomogeneity have been obtained with the help of finite element package Comsol Multiphysics 5.0. It has been shown that the inhomogeneity of the plate have a profound impact on the first (lowest) frequency of the plate, while the plate orthotropy has a greater influence on the second and higher vibration mode [2] (Fig. 1, Table 1).

On the Unsymmetrical Buckling of the Nonuniform Orthotropic Circular Plates

This work is concerned with the numerical study of unsymmetrical buckling of clamped orthotropic plates under uniform pressure. The effect of material heterogeneity on the buckling load is examined. The refined 2D shell theory is employed to obtain the governing equations for buckling of a clamped circular shell. The unsymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value, which leads to the appearance of waves in the circumferential direction. It is shown that if the elasticity modulus decreases away from the center of a plate, the critical pressure for unsymmetric buckling is sufficiently lower than for a plate with constant mechanical properties.

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.

Sensitivity analysis of mathematical models of the IOP changes

The present paper is concerned with study of the effects of different characteristics of the corneoscleral shell of the human eye on intraocular pressure (IOP) changes after intravitreal injections. Sensitivity analysis was performed to quantify the relative importance of material and geometrical parameters of the eyeball on the IOP elevation.

Application of shell theories for simulation of intraocular pressure changes after injection

The stress-strain state of a pressurised spherical shell is studied by means of the exact 3D theory of elasticity and the 2D approximate shell theories of moderate thicknesses. The sphere is made of transversely isotropic material. The problem models the changes in the fluid pressure inside the human eye due to injected additional volume of liquid. The algebraic relationships for deflections and stresses are derived. Described shell theories take into account the variation of the shell thickness, which can be important for soft materials, e.g. the human eye’s tissue. The asymptotical analysis of the exact 3D solutions has been performed and the accuracy of the approximate solutions, obtained with the approximate theories is analyzed. Analytical results found with the presented theories are compared with those obtained with FEM.

Computer simulation of the cornea-scleral shell as applied to pressure-volume relationship in the human eye

Relationship between intraocular pressure (IOP) and the intraocular volume (IOV) was obtained for three mechanical models with following problem statements: 1) the eyeball shell is assumed to be ellipsoidal transversal isotropic shell; 2) the corneoscleral shell of the eye is modeled as a conjugated shell consisting of two segments (sclera and cornea); 3) in the third model the coupled fluid-structure interactions in the eye are taken into account.