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The principal object of this volume is the creation of a
mathematical theory of deformations for elastic anisotropic
thermodynamic piezoelastic plates, beams and shells with variable
thickness. The book is divided into two parts. The first part deals
with problems related to the construction of refined theories (such
as those of Richhof-Love, von Karman-A. Fioppl, and Reissner) and
their equivalent new models (depending on arbitrary control
functions). These are investigated by means of a new variational
principle. Methods of reduction, containing regular processes of
study of spatial problems, are also studied. Topics treated include
problems of solvability, error estimations, convergence of
processes in Sobolev spaces and construction of effective schemes
of solutions of two-dimensional boundary value problems for systems
of partial differential equations. The second part considers stable
projective methods, using classical orthogonal polynomials and a
new class of spline-functions as coordinate systems, and their
numerical realizations for a design of one- and two- dimensional
boundary value problems from the first part. These efficient
methods increase the possibilities of classical finite-difference,
exponential- fitted, variational-discrete and alternating-direction
methods. Audience: This book will be of interest to researchers and
graduate students whose work involves mechanics, analysis, numerics
and computation, mathematical modelling and industrial mathematics,
calculus of variations, and design engineering.
The main purpose of this work is construction of the mathematical
theory of elastic plates and shells, by means of which the
investigation of basic boundary value problems of the spatial
theory of elasticity in the case of cylindrical do mains reduces to
the study of two-dimensional boundary value problems (BVP) of
comparatively simple structure. In this respect in sections 2-5
after the introductory material, methods of re duction, known in
the literature as usually being based on simplifying hypotheses,
are studied. Here, in contradiction to classical methods, the
problems, connected with construction of refined theories of
anisotropic nonhomogeneous plates with variable thickness without
the assumption of any physical and geometrical re strictions, are
investigated. The comparative analysis of such reduction methods
was carried out, and, in particular, in section 5, the following
fact was established: the error transition, occuring with
substitution of a two-dimensional model for the initial problem on
the class of assumed solutions is restricted from below. Further,
in section 6, Vekua's method of reduction, containing regular pro
cess of study of three-dimensional problem, is investigated. In
this direction, the problems, connected with solvability,
convergence of processes, and construction of effective algorithms
of approximate solutions are studied."
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