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A. G. Kolpakov

Publications and source records attributed to A. G. Kolpakov.

7 recordsLinked to original sources
arXiv2021

A new type of boundary layers, a representative model, and surface wrinkling in inhomogeneous plates

We find a new type of boundary layers that occur at the top and bottom surfaces of an inhomogeneous plate. The boundary layers of this type never occur in homogeneous plates or plates made of layers of homogeneous materials. The thickness of such a boundary layer is usually less than that of a one single fiber of the plate. Moreover, we introduce a new notion of representative plate as a plate whose stress-strain state in some parts is similar to the stress-strain state in the original plate. In addition, every part of the original plate has a corresponding part in the representative plate. We demonstrate that a three-layered plate can be representative of a plate formed of an arbitrary number of layers. We also demonstrate a wrinkling phenomenon that happens on the top and bottom surfaces of inhomogeneous plates. This effect does not occur in homogeneous plates or plates made of layers of homogeneous materials. Such wrinkling can become a significant adverse effect, and thus should be given utmost attention.

math-ph↗
arXiv2018

Computation of homogenized stiffnesses of thin corrugated plates

The homogenized stiffnesses of a corrugated plate are calculated by solving the periodicity cell problem of the homogenization theory by using two-steps dimension reduction procedure. The three-dimensional periodicity cell problem first reduces to a two-dimensional problem on the plate cross-sections. Then, provided that the plate is thin, the two-dimensional problem reduces to a one-dimensional problem, similar to the problem of curvilinear beam bending. Using the specifics of the arising one-dimensional problem, we construct its solution, depending on several constants. As a result, the solution of the cell problem and the calculation of the homogenized stiffnesses are reduced to solving algebraic equations. It is found that essential calculations are necessary only for finding the tension stiffness in the direction orthogonal to the corrugation profile. The remaining effective stiffnesses are easily calculated by numerically computing integrals of known functions. For some effective stiffnesses exact formulas are obtained.

physics.class-ph↗
arXiv2014

Asymptotic decomposition in the problem of joined elastic plates

In this paper, a method of local perturbations, previously successfully applied to decompose the problem of elasticity in the system of connected thin rods and beams [Kolpakov and Andrianov, 2013], is used to study the asymptotic behaviour of the elasticity problem in connected thin plates. A complete decomposition of the problem, i.e. the separation of the original problem in to the two-dimensional problem of the theory of plates and local problems is proposed. The local problems describe the three-dimensional stress-strain state in the connected plates and can be solved by numerical methods.

physics.class-ph↗
arXiv2013

Interaction of "rigid" quantum systems

We introduce the notion of a "rigid" quantum system as a system with constant relative positions of its nuclei and constant relative distribution of the electrons with respect to the nuclei. In accordance with this definition, a molecule which does not interact with other objects, is a "rigid" quantum system. Molecule is also "rigid" if it interacts with other objects, but the interaction does not change the intrinsic structure of the molecule (or this change can be neglected). Several "rigid" quantum systems interact one with another in the quantum manner. The interaction is ruled by the Schr{ö}dinger equation [1] written for all the particles of the systems under consideration. We consider the case when the external potential is zero.

math-ph↗
arXiv2012

Complementarity problems for electro-neutral charged bodies

Solutions to the complementarity problem constructed in [1], generally, possess non-zero total charge. In natural sciences, bodies possessing non-zero total charge (ions and similar object) are considered as specific objects. Bodies possessing zero total charge (electro-neutral bodies) are considered as general case objects. This paper presents a solution to the complementarity problem for electro-neutral bodies. The solution is constructed under the condition that the volumes of the bodies are small.

math-ph↗
arXiv2012

Complementarity problems for two pairs of charged bodies

We consider an interaction of charged bodies under the following simplified conditions: the distribution of charge over each body is stable; the interaction of bodies is governed by electrical forces only. Physically, these assumptions can be treated as the following decomposition of charges: the structure of each body is assumed to be stable due to inner forces (say, quantum forces [1]), which do not influence the interaction of the bodies; the bodies interact due to the classical electrical forces [2] only. In this model, the role of inner forces is to create a specific stable distribution of the charge over a body. We assume that the charge distribution over a body can be described by the density of the charge. In our model, the distribution of the charge is the property of a body and does not change in the process of the bodies' interaction. For the simplicity we assume that the bodies are similar in the sense of geometry, say, occupy domain $Q$ and have a preferable direction of interaction denoted by $Ox_3$.

math-ph↗