The range of scientific interests of P.A. Zhilin is very wide. It includes fundamental
laws of mechanics, direct tensor calculus, rigid body dynamics, nonlinear rod theory,
general theory of inelastic media, including plasticity, consolidating granular media,
phase transitions, as well as piezoelectricity, ferromagnetism, electrodynamics and
quantum mechanics. At first sight it seems that the investigations are not related
one to another. But this is not so. Let us show a few examples.
Rigid body oscillator and gyrostat on an elastic foundation, which mathematical
description concerns to area of the rigid body dynamics, are used further as
fundamental model when constructing inelastic media theory, piezoelectricity theory,
theory of magnetoelastic materials, and also for derivation of Maxwell's equations.
Methods of description of the spinor motion, based on use of the direct tensor
calculus, are used and developed both for solving rigid body dynamics problems and
for solving nonlinear rod theory problems. The same methods are used when constructing
various continuum models, which take into account rotational degrees of freedom:
shells, rods, plastic materials, consolidating granular materials, piezoelectric
materials, magnetoelastic materials, and electromagnetic field.
The symmetry theory and tensor invariant theory are being actively used and developed
when constructing shell theory, rod theory, as well as for other continuum theories:
general theory of inelastic media, piezoelectricity theory and ferromagnetism theory.
The huge attention is given base concepts and fundamental laws of mechanics. Some
papers are devoted to a consecutive statement of fundamental laws of the Eulerian
mechanics mechanics of bodies of a general kind, consisting of particles with
rotational degrees of freedom. All continuum theories developed by P.A.Zhilin,
including models, which mathematical description, finally, is reduced to Maxwell's
equations, Schrödinger's equation and Klein-Gordon's equation, are constructed
adhering to the same positions based on the fundamental laws of mechanics.
When constructing continuum models both for elastic (shells, rods, piezoelectric
materials, magnetoelastic materials, electromagnetic field) and inelastic media,
the theory of strains, proposed by P.A. Zhilin, is used. The theory of strains is
based on the idea of using the reduced energy balance equation for defining measures
of deformation.
By the elementary examples of discrete systems mechanics the concept of internal
energy, chemical potential, temperature and entropy are introduced. Definition of
these quantities is given by means of pure mechanical arguments, which are based on
using special mathematical formulation of energy balance equation. The same method
of introducing the basic concepts of thermodynamics is used when constructing
continuum theories.
In fact P.A. Zhilin's papers represent the method for constructing continuum theories
with rotational degrees of freedom together with the necessary mathematical technique,
as well as examples of using the mentioned theories when describing different physical
phenomena, including phenomena which are considered not subject to classical mechanics.
|
![membrane](img/membrane_e.gif)
![rod](img/rod_e.gif)
![dynamics](img/dynamics_e.gif)
![fundamental](img/fundamental_e.gif)
![inelastic](img/inelastic_e.gif)
![electrodynamics](img/electrodynamics_e.gif)
![quantum](img/quantum_e.gif)
![pezo](img/pezo_e.gif)
|