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: 10.18698/2541-8009-2019-1-433
Category: Mechanics | Chapter: Mechanics of Deformable Solid Body
Keywords: shadow finite element, flexible rod, deformation, Hooke’s law, stress-strain 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 tension-compression 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.

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