Finite element formulation of slender structures with shear deformation based on the Cosserat theory

Abstract

This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.

Variational principle for a special Cosserat rod

“…These results agree well with those derived by Newton's dynamical laws. This method has been applied to model a MEMS device Tang resonator [17][18] .…”

“…Sansour and Skatulla [14][15] discussed some aspects of non-linear formulation of the Cosserat continuum by using a modified variational principle and a first strain gradient formulation is derived, which features a generalized deformation description. Cao et al [16] and Liu et al [17] developed a Cosserat rod element and modified Cosserat rod element modelling strategy to model slender structures in the certain dynamical system.…”

Application of the Cosserat continua to numerical studies on the properties of the materials

“…Recently, Rubin, Liu et al, Riahi and Curran, Riahi et al [23][24][25][26] contributed to the numerical solution of structures such as shells, cantilever beams and plates based on three dimensional Cosserat theories. In these researches, the Cosserat model falls within the category of equivalent continuum approaches in which the effects of the layer interfaces and the layer thickness are incorporated in the governing equations and the constitutive formulation and the mesh discretization is not governed by the orientation and the spacing of discontinuities, which becomes appealing from both a mathematical and a practical point of view.…”

Cosserat Rods and Their Application to Drill-String Dynamics

“…The rotation behaviour, the resultant forces and torques in the rod, as well as the contact reaction of the inner surface of the tube, are determined and the differences between the static and dynamic solutions are shown. Lastly, [6] uses a Cosserat rod approach based on the modified Cosserat rod element (MCRE) of [33], but the model is not exploited by analysing curved borehole configurations.…”

“…In the study, attention was paid to the modelling of the drill-pipes section. The implementation of the Cosserat rod in [6] is based on the Modified Cosserat Rod Element (MCRE) approach described by [33,34]. Nevertheless, it is indicated in [35], another researcher from the same group at Aberdeen, that the approach presented in [6,33,34], based on the MCRE, is unsuitable for modelling systems that involve large displacements.…”