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Some problem s of linear and nonlinear theory of shallow and nonshallow shells (FR /358/5-109/14)


Funded by

SRNSFGShota Rustaveli National Science Foundation of Georgia

Start Date: 2015-05-05       End Date: 2018-05-05

By means of Vekua’s method, the system of differential equations for the geometrically and physically nonlinear theory non-shallow shells is obtained. Using the method of a small parameter, by means of Muskhelishvili and Vekua-Bitsadze methods, for any approximations of order N the complex representations of the general solutions are obtained.
I. Vekua obtained the conditions for the existence of the neutral surface of a shell, when the neutral surface is the middle surface. The neutral surface is considered as any equidistant surfaces of the shell.
Wide class of shell type bodies of toroidal stucture with smooth boundary and different basic lines was studied. For middle surfaces of shell bodies of hollow helix type first and second quadratic forms, Gauss and normal curvature will be calculated.
we consider a plane problem of elasticity for a polygonal domain with a curvilinear hole, which is composed of the rectilinear segment (parallel to the abscissa axis) and arc of the circumference and the problem of finding a partially unknown boundary of the plane theory of elasticity for a rectangular domain which is weakened by an equally strong contour (the unknown part of the boundary). The problem is solved by the methods of conformal mappings and boundary value problems of analytic functions. The sought complex potentials are constructed effectively (in the analytical form). Estimates of the obtained solutions are derived in the neighborhood of angular points.
We consider the three-dimensional system of the equations of elastic static equilibrium of bodies with double porosity. From this system of the equations, using a method of a reduction of I. Vekua, we receive the equilibrium equations for the shallow shells having double porosity. Further we consider a case of plates of constant thickness in more detail. Namely, the system of the equations corresponding to approximations N = 1 it is written down in a complex form and we express the general solution of these systems through analytic functions of complex variable and solutions of the Helmholtz equation. The received general representations of decisions give the opportunity to analytically solve boundary value problems about elastic equilibrium of plates with double porosity.

Project members:

Publications

  • Bakur Gulua, The Method of I. Vekua for the Non-Shallow Spherical Shell for the Geometrically Nonlinear Theory, AMIM, 20 (2), 3-9, Tbilisi University Press, 2015.
  • Giorgi Kapanadze, Bakur Gulua, A Problem of Plane Elasticity for a Rectangular Domain with a Curvilinear Quadrangular Hole, AMIM, 20 (2), 24-33, Tbilisi University Press, 2015.
  • Bakur Gulua, Normed Moments Method for Non-Shallow Shells, AMIM, 20 (1), 13-20, Tbilisi University Press, 2015.
  • Tengiz Meunargia, Generalization of I. Vekua Reduction Method for Physically and Geometrically Non-Linear and Non-Shallow Shells, AMIM, 20 (1), 36-46, Tbilisi University Press, 2015.
  • Tengiz Meunargia, Neutral surfaces of a non-shallow shells, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, Volume 29, 88-91, Tbilisi University Press, 2015.
  • Bakur Gulua, One boundary value problem for the plates, Seminar of I. Vekua Institute of Applied Mathematics REPORTS, Volume 42, 3-9, Tbilisi University Press, 2016.
  • Giorgi Kapanadze, Bakur Gulua, One problem of the bending of a plate for a curvilinear quadrangular domain with a rectilinear cut, Seminar of I. Vekua Institute of Applied Mathematics REPORTS Volume 42, 27-33, Tbilisi University Press, 2016.
  • Tengiz Meunargia, The isometric system of coordinates and the complex form of the system of equations for the non-shallow and nonlinear theory of shells, Seminar of I. Vekua Institute of Applied Mathematics REPORTS, Volume 42, 47-53, Tbilisi University Press, 2016.
  • Giorgi Kapanadze, Bakur Gulua, About One Problem of Plane Elasticity for a Polygonal Domain with a Curvilinear Hole, AMIM, 21 (1), 121-129, Tbilisi University Press, 2016.
  • Bakur Gulua, Solution of Boundary Value Problems of Spherical Shells by the Vekua Method for Approximation N=2, AMIM, 21 (2), 3-15, Tbilisi University Press, 2016.
  • Bakur Gulua, Giorgi Kapanadze, Some Boundary Value Problems for Plane Theory of Elasticity for Doubly-Connected Domain, AMIM, 21 (2), 38-45, Tbilisi University Press, 2016.
  • Bakur Gulua, Roman Janjgava, Miranda Narmania, Derivation of System of the Equations of Equilibrium for Shallow Shells and Plates, Having Double Porosity, AMIM, 21 (2), 16-37, Tbilisi University Press, 2016.
  • Tengiz Meunargia, On the 2-D Nonlinear Systems of Equations for Non-Shallow Shells (E. Reissner, D. Naghdi, W. Koiter, A. Lurie, I. Vekua), AMIM, 22 (2), 64-72, Tbilisi University Press, 2016.
  • Bakur Gulua, On a boundary value problem for the nonlinear non-shallow spherical shell, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, Volume 30, 23-26, Tbilisi University Press, 2016.
  • Tengiz Meunargia, The problem of existence the neutral surface for the elastic shell, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, Volume 30, 74-77, Tbilisi University Press, 2016.
  • Roman Janjgava, Bakur Gulua, Miranda Narmania, The Boundary Value Problem of Plates with Double Porosity by the Vekua Method for Approximations N=1, AMIM (Applied Mathematics, Informatics and Mechanics) 22(1), 2017, 50-57, Tbilisi University Press , 2017.
  • Roman Janjgava, Bakur Gulua, The Dirichlet Boundary Value Problem of Porous Cosserat Media with Triple-porosity for the Concentric Circular Ring, AMIM 22 (1), 42-49, Tbilisi University Press , 2017.
  • Bakur Gulua, Roman Janjgava, Some basic boundary value problems for plane theory of elasticity of porous Cosserat media with triple‐porosity, PAMM·Proc.Appl.Math.Mech.17, 705–706, Wiley, 2017.
  • Bakur Gulua, Roman Janjgava, Tamar Kasrashvili, One problem of porous Cosserat media for solids with triple-porosity, AMIM, 22 (2), 3-15, Tbilisi University Press, 2017.
  • Giorgi Kapanadze, The problem of finding an equally strong contour for a rectangular plate weakened by a rectilinear cut, whose ends are cut out by convex smooth arcs, Proceedings of I. Vekua Institute of Applied Mathematics Volume 67, 69-75, Tbilisi University Press, 2017.
  • Bakur Gulua, On one boundary value problems for a circular ring with double porosity, Proceedings of I. Vekua Institute of Applied Mathematics Volume 67, 34-40, Tbilisi University Press, 2017.
  • Bakur Gulua, Roman Janjgava, The basic boundary value problem for the plane theory of elasticity of porous Cosserat media with triple-porosity, Proceedings of I. Vekua Institute of Applied Mathematics Volume 67, 41-50, Tbilisi University Press, 2017.
  • Tengiz Meunargia, On the imbedding of the surface in the 3-D Riemannian manifold, Proceedings of I. Vekua Institute of Applied Mathematics Volume 67, 94-99, Tbilisi University Press, 2017.
  • Bakur Gulua, Basic boundary value problems for circle with double porosity, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, Volume 31, 51-54, Tbilisi University Press, 2017.
  • Bakur Gulua, Roman Janjgava, Boundary value problems of the theory of elasticity of porous Cosserat media for solids with triple-porosity, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, Volume 31, 55-58, Tbilisi University Press, 2017.
  • Bakur Gulua, Tamar Kasrashvili, On One Problem for the Plate, AMIM, 24 (1), 23-30, Tbilisi University Press, 2019.
  • Bakur Gulua, Roman Janjgava, Tamar Kasrashvili, Miranda Narmania, Some BVP in the plane theory of thermodynamics with microtemperatures, AMIM, 24 (2), 10-19, Tbilisi University Press, 2019.
  • Tengiz Meunargia, On the nonlinear theory of non-shallow shells, AMIM, 24 (2), 35-50, Tbilisi University Press, 2019.
  • Giorgi Akhalaia, Tengiz Meunargia, Conditions for the existence of neutral surface of an elastic shell and the boundary value problems for generalized analytic functions, AMIM, 24 (1), 3-13, Tbilisi University Press, 2019.