Tension of a strip made of a compressible material with a continuous velocity field under plane deformation
DOI: 10.34759/trd-2022-124-07
Аuthors
1*, 1**, 2***, 1****1. Komsomolsk-na-Amure State University, 27, Lenina str., Komsomolsk-on-Amur, 681013, Russia
2. Institute of Applied Mathematics, Far Eastern Branch of the Russian Academy of Sciences, Vladivostok, Russia
*e-mail: naj198282@mail.ru
**e-mail: Grigoreva.al@email.knastu.ru
***e-mail: khromovai@list.ru
****e-mail: fct@knastu.ru
Abstract
The subject of the study of the presented article consists in determining conditions of the moment of a crack nucleation at the stretching of the plane sample from the compressible material under conditions of the plane deformation.
The purpose of this study is studying the process of stretching a flat sample made from compressible material with a continuous displacement velocities field under the plain strain conditions. The material compressibility associated with the mass conservation law, which is formulated in the form of the equation of continuity, leads to a density change while loading process, in accordance with which the logarithm of the material derivative of density in time is being added to the system of equations defining the displacement velocity field. This system is being converted to a system of inhomogeneous wave equations.
The following methods were applied while this research conducting. They are:
— Analytical method for the velocity and deformations field determining;
— Averaging method for obtaining the mean value of the auxiliary function on the sphere;
— Method of descent for transition from the sphere surface integration to the circle integration.
In the course of the studies, an intermediate result, namely the general solution of the system of wave equations for the velocities field determining, was obtained.
The results of the study can be applied while mathematical models developing of the behavior of real structures’ elements in the problems of modern mechanical engineering and construction, as well as in assessing their strength.
Keywords:
plane deformation, rigid-plastic body, material compressibility, density, system of wave equationsReferences
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