Математична модель деформування тонкої в’язкопружної платівки

О. М. Нещадим, О. П. Зінькевич, В. М. Сафонов

Анотація


MATHEMATICAL MODEL OF DEFORMATION

THIN VISCOELASTIC PLATE

A. Neshchadym,  A. Zinkevych, V. Safonov

 

The need to develop effective methods for solving boundary value problems viscoelasticity dictated many practical applications of such problems in engineering and physics. Engineers and researchers interested in convenient mathematical methods that allow through numerical calculations to simulate simple patterns. Among these methods advantageously distinguished boundary integral equation method, which is easy to algorithmic.

Methods of potential theory used in the theoretical and practical studies mathematical models of many applied problems of mathematical physics. In most cases the border region is considered unchanged, or one that changes little. This paper develops a method of boundary integral equations for solving planar linear viscoelasticity boundary problem with moving boundary.

         Used fundamental solution of integral-differential equations and discharged integral representation of the desired solution of the boundary problem through viscoelastic potentials. Such integrals containing products of fundamental solution (functions dependent variables and integration parameters - time and coordinates in a given area) for the potential density. It is important that the integrated dependent binding integrals of potentials study area on the border of this area. The substitution of the integral representation of the desired solution of the boundary conditions based viscoelastic properties of potential leads to extremely time-integral equations of the second kind in respect of certain arbitrary functions - density potential. These integral equations containing integrals over the boundary of the domain occupied by the plate, and by the time variable.

  In practice, the numerical simulation of the unknown potential density determined from a system of linear algebraic equations, which replaces approximately obtained integral equations. The algorithm that implements the method of "time steps" solution of the system and finding the coordinates of points

on the moving boundary  of the region .

 


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