Svetlana M. Bauer

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.

Nonclassical Theories of Shells in Application to Soft Biological Tissues

Two non-classical theories for orthotropic plates of moderate thicknesses are discussed. In these theories both deformations, rotation and bending of the fibers and their elongations in the direction of the thickness of the shell are taken into account. The stress-strain state of a circular plate modeling the Lamina Cribrosa in the human eye is studied by means of these theories. Numerical results for displacements and stresses found with the presented theories are compared with those obtained with FEM.

Role of the outer stomatal ledges in the mechanics of guard cell movements

Key MessageThe modelling showed that outer ledges prevent wide opening of the stomatal pore and its lifting above leaf epidermis. This stomatal mechanics is combined with xeromorphic features of leaf epidermis.AbstractMethods of light, scanning, and transmission electron microscopy were used to study the stomata of the leaf epidermis in evergreen Acokanthera oblongifolia (Apocynaceae), A. oppositifolia (Apocynaceae), Carissa spectabilis (Apocynaceae), Exbucklandia populnea (Hamamelidaceae), and Trochodendron aralioides (Trochodendraceae). The stomata of their leaf epidermis are located on subsidiary cells, have large outer ledges, and lack inner ledges. To elucidate the role of the ledges, we applied dynamic modelling using the finite-element method. The application of dynamic modelling has shown that outer ledges prevent wide opening of the stomatal pore and their rising above the surface of leaf epidermis. The results of the modelling are supported by the observed deformations in the guard cells of the real stomata. This stomatal mechanics is combined with such stomatal xeromorphic features as thick cuticle, stomatal cavities, and waxy plugs (in A. oblongifolia). All studied species show similar leaf anatomy. It has much in common with the leaf anatomy of species connected in their origin with subhumid Tertiary laurophyllous forests.

Axisymmetric deformations of the orthotropic spherical layer under normal pressure

The deformation of the elastic transversally isotropic layer under normal pressure in a three-dimensional formulation is analyzed. The asymptotic formulas that describe the behavior of the soft layer, the rigidity of which in the transversal direction is much lower compared with the tangential direction are obtained. A comparison has been carried out of the exact and asymptotic solutions. The influence of the parameters on the value in relation to a change in the layer thickness is examined.

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.

Stability loss in an infinite plate with a circular inclusion under uniaxial tension

Loss of stability under uniaxial tension in an infinite plate with a circular inclusion made of another material is analyzed. The influence exerted by the elastic modulus of the inclusion on the critical load is examined. The minimum eigenvalue corresponding to the first critical load is found by applying the variational principle. The computations are performed in Maple and are compared with results obtained with the finite element method in ANSYS 13.1. The computations show that the instability modes are different when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. As the Young’s modulus of the inclusion approaches that of the plate, the critical load increases substantially. When these moduli coincide, stability loss is not possible.

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).

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.

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.

Intravitreal injection of triamcinolone acetonide and intraocular pressure: author's reply

Editor, T he guest editorial (Wickström 2008) and review article (Høvding 2008) appearing in Acta Ophthalmologica this year discussed the practical problems concerning treatment ⁄delayed treatment ⁄no treatment at all of acute bacterial conjunctivitis well. Høvding (2008) emphasizes the difficulty in making a correct distinction between bacterial and viral conjunctivitis. This can be done with the simple leucocyte–esterase stix test, which has a sensitivity of 89% and a specificity of 98% in bacterial conjunctivitis. In virus conjunctivitis with lymphocytosis it is negative Norn (1989). The tear fluid is transferred from the lateral part of the lower fornix to the stix with the end of a glass rod. The red colour of the stix is judged after exactly 1 min depending on the degree of bacterial infection Norn (1992). I admit that this method, published for practitioners, contact lens optometrists and ophthalmologists, has not been used much until now. But it could be of value in some cases.