Sessions 2011-2012

Various mathematical models of lamina cribrosa (LC) in eye, which describe the behavior of LC if IOP changes and consider such common properties of living tissue as anisotropy and viscoelasticity, are considered.

LC is modeled by transversely isotropic round plate. Results of various theories and ANSYS results under different geometric and mechanical parameters are compared. Also a layered structure of LC is taken into account. Stress-strain state of two-layer plate with different parameters for each layer is compared to stress-strain state of one-layer plate with equal mechanical parameters. The material of LC is considered viscoelastic. The influence of short-term and long-term loadings to stress-strain state of LC is examined.

The deformations of shells in applying a load with a flat base are significant and described by equations of geometrically nonlinear theory of shells. These arch of solutions of nonlinear theory of shells presents certain difficulties. These difficulties reduce dramatically in case of limited displacements under which the form of the shell does not change much in the process of loading. That is why the solution of this problem is based on the method of gradual loadings – delta method. The delta method is characterized by the fact that because of the application of linear physical relations it is possible to minimize the problem at each step to the solving linear system with constant coefficients. The calculation is finished when the loading or deformations reaches the given finite value. For solving the considered problem Paliy-Spiro theory of anisotropic shells of medial thickness is used.

The problems of membrane and soft shells under the load with a flat base are important for the analysis of the data related to measurement of intraocular pressure - the characteristic which is very important for ophthalmology.

The report addresses the problem of modeling the jets flowing out of nozzles engines, with different environmental parameters and compositions of combustion products. Applied significance the problem being solved - an optical identification of the flying vehicles, so it is important to take into account when modeling mixing, burning of, the presence of the dispersed phase. Also singularity the developed algorithm is a wide range of application. A mathematical model based on fundamental conservation laws and well-known empirical dependences.

Flat erected filtration in a ground from sprinklers and furrow soil is considered. In this hydrodynamic statement, capillarity of the ground and transpiration from the free surface are considered. Some models of filtering through a soil stratum with underlying permeable pressure aquifer are presented. The mixed multiparameter boundary-value problem of the theory of analytic functions is stated. It is solved by applying the method of P.Ya. Polubarinova-Kochina. On the basis of this model, the algorithm for calculating a capillary diffluence of water and the filtrational flux rate is developed. Situations when at a filtration of water from sprinklers the ground capillarity is considered, transpiration from the free surface of subsoil waters, and also skids from waters of underlying well pervious layer are viewed. The hydrodynamic analysis of the structure and the prominent features of the modeled process, and also the influence of all physical properties of the system on filtrational performances is carried out. The analysis is done by means of the gained exact analytical dependences and numerical calculations. Limiting and special cases are considered. They are related to the lack a few factors characterizing the modeled process, which have been studied previously by other authors. Computed results are compared at identical filtrational performances with similar plans.

Memoirs in photos by S.B. Filippov, A.L. Smirnov, A.M. Ermakov, A.V. Lebedev, L.A. Karamshina and others.

The problem on an elastic plane weakened by an elliptical hole of a nanometer size with the boundary free from an external traction under arbitrary remote loading is solved. A residual stress and complementary surface stress acting at the boundary are taken into account. Based on Kolosov-Muskhelishvili’s method, the solution of the problem is reduced to the solution of a singular integro-differential equation in the unknown surface stress. For particular case of a circular nanometer hole, this equation is solved in an explicit form. Dependences of the hoop stresses at the boundary on the hole radius is obtained with and without the residual stress and the complementary surface stress. As it follows from this solution, the stress concentration depends on the elastic properties of a surface and bulk material, and the size of a hole as well.

Several buckling problems for plates and shells modelled with non-classical theories are discussed in the thesis :
1. Buckling of axially compressed plate in the channel with rigid walls.
2. Buckling of transversely isotropic spherical shell under normal pressure.
3. Buckling of transversely isotropic cylindrical shells under axial compression.

A mathematical model of deformation of shallow shells of rectangular plan, supported by the ribs of variable height. A variant of the method of structural anisotropy, which reduces the discrete shell of variable thickness of the envelope of continuously variable thickness. Based on the method of continuation of solution for the best parameter algorithm of research supported by the stability of shallow shells and the software implementation of this algorithm. A study of the stress-strain state, stability and supercritical behavior of shallow shells supported by ribs, fixed and variable height. The efficiency of the reinforcement of the shell edges of variable height.

A mathematical model of deformation of reinforced shells of revolution with linear and nonlinear elastic deformation and creep in the light of the material. An algorithm for the study of this model based on the functional use of the total energy of the deformation gradient method. The software complex investigation of strength and stability of reinforced shells of rotation and the results of an investigation for a variety of shells of revolution.

A problem of stress concentration in vicinity of edge of joint between the tile and the base is considered. The singular state of stress and strain is described by asymptotic formulas, which are defined by geometry of vicinity of the joint and material properties. An information about the loads applied far from the joint is concentrated in scaling factor — stress intensity factor (SIF).

Known in fracture mechanics force strength criterion is being used to estimate the strength of the joint, which requires SIF to be in the range of admissible values in the case of the problem of an edge of joint.

A method of SIF numerical calculation based on reciprocal theorem usage in vicinity of the edge is being introduced. The method allows generalization to the case of thermal loads.

Work is devoted to numerical research of pressure waves induced by motion of a high-speed train in a tunnel by means of package "ANSYS Fluent".
The effect on the pressure gradient on
- the length of a tunnel;
- the ratio of the cross-section areas of a train and a tunnel;
- the form of a head part of a train;
- the speed of a train
has been studied.
The graphics for the resistance and elevating forces have been plotted.