Some problems for elastic double-layer bodies

Author: Nino Bliadze
Keywords: problems, elastic, double-layer, bodies
Annotation:

In the fifties of XX century investigations of cusped elastic prismatic shells actually takes its origin, namely, in 1955 I.Vekua raised the problem of investigation of elastic cusped prismatic shells, whose thickness on the prismatic shell entire boundary or on its part vanishes.In practice, such cusped prismatic shells, in particular, cusped plates, and cusped beams (i.e.,beams whose cross-sections area vanishes at least at one end of the beam) are often encountered in spatial structures with partly fixed edges, e.g., stadium ceilings, aircraft wings, submarine wings etc., in machine-tool design, as in cutting-machines, planning-machines, in astronautics, turbines, and in many other application fields of engineering. Investigation of elastic cusped prismatic shells, considered as 3D ones, may occupy 3D domains with, in general, non-Lipschitz boundaries. The problem mathematically leads to the question of setting and solving of boundary value problems for even order equations and systems of elliptic type with the order degeneration in the static case and of initial boundary value problems for even order equations and systems of hyperbolic type with the order degeneration in the dynamical case. At the same time I.Vekua introduced a new mathematical model for elastic prismatic shells which was based on expansions of the three-dimensional displacement vector fields and the strain and stress tensors in linear elasticity into orthogonal Fourier-Legendre series with respect to the variable plate thickness. By taking only the first N + 1 terms of the expansions, he introduced the so called N-th approximation. Each of these approximations for N = 0; 1; ... can be considered as an independent mathematical model of plates. In particular, the approximation for N = 1 corresponds to the classical Kirchhoff plate model. In the sixties, I. Vekua developed the analogous mathematical model for thin shallow shells. Works of I. Babuska, D. Gordeziani, T. Vashakmadze, V. Guliaev, I. Khoma, A. Khvoles, T. Meunargia, C. Schwab, V. Zhgenti, G. Jaiani, G. Tsikarishvili, M. and G. Avalishvili, W. Wendland, D. Natroshvili, S. Kharibegashvili, N. Chinchaladze, R. Gilbert, and others are devoted to further analysis of I.Vekua's models (rigorous estimation of the modeling error, numerical solutions, etc.) and their generalizations . The analogues system in the case of (N3,N2) approximation of hierarchical models for cusped beams, in general, beams with variable rectangular cross-sections are derived by G. Jaiani . His goal was to find out the peculiarities of setting the boundary conditions for primarily cusped bars and to determine its geometric-mechanical seuse. In the present work consider a two-layer prismatic shell consisting of two cusped elastic bars. In introduction is devoted to review of the literature. The first chapter provides auxiliary material, and the second chapter deals with the main results of the master's thesis.