Gradient model of bending of a composite beam

Authors

  • Vatulyan A.O. Southern Federal University, Rostov-on-Don, Russian Federation ORCID 0000-0003-0444-4496
  • Nesterov S.A. Southern Mathematical Institute, Vladikavkaz Scientific Center of Russian Academy of Sciences, Vladikavkaz, Russian Federation ORCID 0000-0003-3780-5104

UDC

539.3

DOI:

https://doi.org/10.31429/vestnik-19-2-6-16

Abstract

Size-dependent models of beam bending have attracted increased attention of scientists in recent years. The gradient theory of elasticity is used for the correct description of experimental data for micro-dimensional beams. However, the problem of obtaining simplified analytical expressions for moments and deflections of a composite beam remains unexplored. A study of the problem of bending a composite Euler-Bernoulli beam taking into account large-scale effects is carried out. To account for the size effects, the one-parameter gradient model of Aifantis is used. One end of the beam is rigidly fixed. Three types of beam loading are considered: 1) evenly distributed transverse force; 2) bending moment; 3) transverse force at the other end of the beam. Within the framework of the gradient theory of elasticity, the problem is formulated in terms of the bending moment, while additional boundary conditions and conjugation conditions are set. Bending moments are represented as the sum of solutions of the problem in the classical formulation and additional gradient terms. Simplified asymptotic expressions for gradient terms for small values of the scale parameter are obtained for each type of load. The limits of applicability of the asymptotic approach are investigated. Analytical formulas for finding the deflection of the middle line of the beam under arbitrary laws of inhomogeneity of bending stiffness are obtained. Calculations of moments and deflection, both in the case of homogeneous and inhomogeneous parts of the beam, are carried out on specific examples. The following was found out: bending moments experience a jump on the interface surface; deflections are continuous; an increase in the scale parameter leads to a decrease in the deflection value. The dependence of the moment jump on the flexural stiffness modules and the scale parameter is investigated. A comparative study of the influence of the value of the inhomogeneity parameter on the deflection distribution was carried out.

Keywords:

gradient theory of elasticity, bending moment, composite beam, Euler-Bernoulli model, asymptotic solution, inhomogeneous materials

Acknowledgement

This work was supported by the grant from the Russian Science Foundation (project n0. 22-11-00265)

Author Infos

Alexander O. Vatulyan

д-р физ.-мат. наук, заведующий  кафедрой теории упругости Института математики, механики и компьютерных наук им. И.И. Воровича Южного федерального университета

e-mail: aovatulyan@sfedu.ru

Sergey A. Nesterov

канд. физ.-мат. наук, старший научный сотрудник отдела дифференциальных уравнений Южного математического института – филиала ВНЦ РАН

e-mail: 1079@list.ru

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Issue

2022. Vol. 19. No. 2

Section

Mechanics

Pages

6-16

Submitted

2022-06-10

Published

2022-06-30

How to Cite

Vatulyan A.O., Nesterov S.A. Gradient model of bending of a composite beam. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2022, vol. 19, no. 2, pp. 6-16. DOI: https://doi.org/10.31429/vestnik-19-2-6-16 (In Russian)

Copyright (c) 2022 Vatulyan A.O., Nesterov S.A.

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