Models of Solid Mechanics in the Problems of Ophthalmology

IF 0.4 Q4 MATHEMATICS

Vestnik St Petersburg University-Mathematics Pub Date : 2023-12-01 DOI:10.1134/s1063454123040039

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Abstract

This paper presents a very brief review of models constructed in cooperation with ophthalmologists, namely, for the change in the stress-strain state of the eye membrane after vision-correction operations and the change in the intraocular pressure after the injection of drugs into the vitreous body. The mathematical models describing the process of measuring the true intraocular pressure (IOP) by applanation methods are discussed. The models of eye biomechanics provide the possibility to obtain a series of new results in solid mechanics, i.e., to solve the problem of the stability of a spherical shell under a concentrated force and inner pressure and the stability of an axisymmetric equilibrium state of inhomogeneous annular orthotropic plates under a uniformly distributed normal load. The problems of the deformation of transversally isotropic spherical and cylindrical layers under inner and outer pressure are analyzed and the solutions obtained within the framework of the three-dimensional theory are compared with those found by the non-classical theories of shells. This comparison makes it possible to estimate the precision of some theories.

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眼科问题中的固体力学模型

摘要本文简要回顾了与眼科医生合作建立的模型，即视力矫正手术后眼膜应力应变状态的变化模型和向玻璃体注射药物后眼压变化模型。本文讨论了描述用applanation方法测量真实眼压（IOP）过程的数学模型。眼球生物力学模型为获得一系列固体力学新成果提供了可能，即解决球壳在集中力和内压作用下的稳定性问题，以及非均质环状正交板在均匀分布法向载荷作用下的轴对称平衡状态的稳定性问题。分析了横向各向同性球形层和圆柱形层在内外压力作用下的变形问题，并将三维理论框架内获得的解与非经典壳理论获得的解进行了比较。通过比较，可以估算出某些理论的精确度。

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来源期刊

Vestnik St Petersburg University-Mathematics MATHEMATICS-

CiteScore

0.70

自引率

50.00%

发文量

期刊介绍： Vestnik St. Petersburg University, Mathematics is a journal that publishes original contributions in all areas of fundamental and applied mathematics. It is the prime outlet for the findings of scientists from the Faculty of Mathematics and Mechanics of St. Petersburg State University. Articles of the journal cover the major areas of fundamental and applied mathematics. The following are the main subject headings: Mathematical Analysis; Higher Algebra and Numbers Theory; Higher Geometry; Differential Equations; Mathematical Physics; Computational Mathematics and Numerical Analysis; Statistical Simulation; Theoretical Cybernetics; Game Theory; Operations Research; Theory of Probability and Mathematical Statistics, and Mathematical Problems of Mechanics and Astronomy.