Implementation of the shadow finite element method for solving problems of deformation statics of curvilinear flexible rods
Authors: Tsaplin I.A., Kiryukhin A.A.  
Published in issue: #1(30)/2019  
DOI: 10.18698/2541800920191433  
Category: Mechanics  Chapter: Mechanics of Deformable Solid Body 

Keywords: shadow finite element, flexible rod, deformation, Hooke’s law, stressstrain state, Newton method, Lagrange principle, full potential of an elastic system 

Published: 04.02.2019 
The paper is concerned with the a finite element model of a flexible rod solving flat problems of statics of thin rods in the case of large displacements and rotations which allows to implement the shadow element method. The deformed state of the final element is represented as a superposition of tensioncompression and bending. Each of the two nodes of the finite element has three degrees of freedom: two linear displacements and rotation. The authors calculated the energy of deformations of the element by small relative displacements and rotations of the nodes, which were separated from full displacements and rotations. Unknown nodal displacements were determined using direct minimization of the full potential of the rod model. In this paper, the authors used the Newton method in the Wolfram Mathematic software package to the search for the minimum. The developed final shadow element is tested on the tasks of deforming curvilinear rods. In conclusion, the paper points out that the comparison of the obtained results with known solutions of the same problems using differential equations confirmed the effectiveness of the method being implemented.
References
[1] Popov E.P. Teoriya i raschet gibkikh uprugikh sterzhney [Theory and calculation of elastic rods]. Moscow, Nauka Publ., 1986 (in Russ.).
[2] Popov V.V., Sorokin F.D., Ivannikov V.V. A flexible rod finite element with separate storage of cumulated and extra rotations for large displacements of aircraft structural parts modeling. Trudy MAI, 2017, no. 92. URL: http://www.trudymai.ru/published.php?ID=76832&eng=N (in Russ.).
[3] Bakhvalov N.S., Zhidkov N.P., Kobel’kov G.M. Chislennye metody [Numerical methods]. Moscow, BINOM. Laboratoriya znaniy Publ., 2015 (in Russ.).
[4] Maklakov S.F. Raschet sterzhnevykh sistem metodom konechnykh elementov [Calculation of rod systems by finite elements method]. RostovnaDonu, RSTU Publ., 2008 (in Russ.).
[5] Svetlitskiy V.A. Mekhanika sterzhney. Ch. 1 Statika [Rod mechanics. P. 1. Statics]. Moscow, Vysshaya shkola Publ., 1987 (in Russ.).
[6] Gavryushin S.S. Vychislitel’naya mekhanika [Computational mechanics]. Moscow, Bauman MSTU Publ., 2017 (in Russ.).
[7] Ignat’yev A.V. Main formulations of the finite element method for the problems of structural mechanics. Part 3. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering], 2015, no. 1, pp. 16–26 (in Russ.).
[8] D’yakonov V.P. Mathematica 5.1/5.2/6. Programmirovanie i matematicheskie vychisleniya [Mathematica 5.1/5.2/6. Programming and mathematical calculations]. Moscow, DMKPress Publ., 2008 (in Russ.).
[9] Levin V.E., Pustovoy N.E. Mekhanika deformirovannykh krivolineynykh sterzhney [Mechanics of curved bar deformation]. Novosibirsk, NGTU Publ., 2008 (in Russ.).
[10] Zienkiewiez O.C. The finite element method in engineering science. McGrawHill, 1971. (Russ. ed.: Metod konechnykh elementov v tekhnike. Moscow, Mir Publ., 1975.)
[11] Ponomarev S.D. Andreeva L.E. Raschet uprugikh elementov mashin i priborov [Calculation of elastic elements for machines and devices]. Moscow, Mashinostroenie Publ., 1980 (in Russ.).
[12] Fokin V.G. Metod konechnykh elementov v mekhanike deformiruemogo tverdogo tela [Finite elements method in deformable body mechanics]. Samara, SamGTU Publ., 2010 (in Russ.).