George Jaiani

On a Mathematical Model of Bars with Variable Rectangular Cross-sections

Der Vekuasche Ansatz [1] des Aufbaus von Platten- und Schalentheorien, bei dem die Felder der Verschiebungen, der Deformationen und der Spannungen des dreidimensionalen Modells der linearen Elastizitatstheorie in Fourier-Legendresche Reihen nach der Veranderlichen der Dicke entwickelt werden, wird zum Aufbau einer Stabtheorie verallgemeinert. Diese Grosen werden hierbei in doppelte Fourier-Legendresche Reihen nach den Veranderlichen Dicke und Breite entwickelt. Sodann werden alle auser den ersten (N3 + 1) (N2 + 1), N3, N2 = 0, 1, … , Gliedern vernachlassigt. Eine solche Naherung des dreidimensionalen Modells durch ein eindimensionales Modell heist (N3, N2)-Approximation. Die Frage der Wohlgestelltheit der Anfangs- und Randwertprobleme wird untersucht. Der Fall, in dem der veranderliche Querschnitt zu einem Intervall oder einem Punkt entartet, wird auch betrachtet. Solche Stabe heisen zugespitzte Stabe (s. auch [2]). We generalize an idea of I. Vekua [1] who, in order to construct a theory of plates and shells, expands the fields of displacements, strains, and stresses of the three-dimensional theory of linear elasticity into orthogonal Fourier-Legendre series with respect to the variable thickness, to a bar model. In the bar model all above-mentioned quantities are expanded into orthogonal double Fourier-Legendre series with respect to the variables thickness and width of the bar, and then all but the first (N3 + 1) (N2 + 1), N3, N2 = 0, 1, …, terms are neglected. This case is called (N3, N2) approximation. The question of well-posedness of the initial and boundary value problems is investigated. The cases in which a variable cross-section degenerates to a segment of a straight line or into a point are also considered. Such bars are called cusped bars (see also [2]).

On Cusped Shell-like Structures

This paper is updated concise survey of results concerning elastic cusped shells, plates, and beams and cusped prismatic shell-fluid interaction problems.

Initial and Boundary Value Problems for Singular Differential Equations and Applications to the Theory of Cusped Bars and Plates

This paper gives an up-to-date overview of the theory of cusped bars, plates, and shells, and related problems of partial differential equations with order degeneration.

On a nonlocal boundary value problem for a system of singular differential equations

This article deals with a system consisting of singular partial differential equations of the first and second order arising in the zero approximation of I. Vekuas hierarchical models of prismatic shells, when the thickness of the shell varies as a power function of one argument and vanishes at the cusped edge of the shell. For this system of special type a nonlocal boundary value problem in a half-plane is solved in the explicit form. The boundary value problem under consideration corresponds to stress–strain state of the cusped prismatic shell under the action of concentrated forces and moments.

Cusped elastic beams under the action of stresses and concentrated forces

A dynamical problem in the (0, 0) approximation of elastic cusped prismatic beams is investigated when stresses are applied at the face surfaces and the ends of the beam. Two types of cusped ends are considered when the beam cross-section turns into either a point or a straight line segment. Correspondingly, at the cusped end either a force concentrated at the point or forces concentrated along the straight line segment is applied. We prove the exists and uniqueness theorems in appropriate weighted Sobolev spaces.

On Physical and Mathematical Moments and the Setting of Boundary Conditions for Cusped Prismatic Shells and Beams

This paper deals with the analysis of the physical and geometrical sense of N-th (N=0,1,…) order moments and weighted moments of the stress tensor and the displacement vector in the theory of cusped prismatic shells [1,2] and beams [3]. The peculiarities of the setting of boundary conditions at cusped edges in terms of moments and weighted moments are analyzed. The relation of such boundary conditions to the boundary conditions of the three-dimensional theory of elasticity is also discussed.

On a generalization of the Keldysh theorem

The Keldysh theorem for an elliptic equation with characteristic parabolic degeneration is generalized for the case of an elliptic equation of the second-order canonical form with order and type degeneration. The criteria under which the Dirichlet or Keldysh problems are correct are given in a one-sided neighborhood of the degeneration segment, enabling one to write the criteria in a single form. Moreover, some cases are pointed out in which it is even nessesary to give a criterion in the neighborhood because it is impossible to establish it on the segment of degeneracy of the equation.

Basic problems of thermoelasticity with microtemperatures for the half-space

Abstract The present paper deals with the three-dimensional linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. In the half-space solutions of some basic boundary value problems are constructed in quadratures.

Cusped Shells and Plates

The present chapter is devoted to a survey concerning BVPs for elastic cusped shells, prismatic shells and plates. Researches are carried out within the framework of hierarchical models and classical bending models. Cusped orthotropic plates and cusped plates on an elastic foundation are explored. Vibration problems are studied. Internal concentrated contact interactions in elastic cusped prismatic shell-like bodies are also analyzed. Fundamental conclusions characterizing peculiarities depending on geometry of sharpening of cusped edges are made. \(N=0\) approximation with plane stress, generalized plane stress, and plane deformation, and \(N=1\) approximation with Kirchoff-Love plate model are compared.

Cusped Prismatic Shell–Fluid Interaction Problems

Only a few papers are devoted to cusped elastic structure–fluid interaction problems. In this chapter such works are surveyed when in the solid part either Kirchoff-Love plate or Vekua’s zero approximation model and in the fluid part either incompressible ideal or viscous fluids are considered.