Finite Element Analysis of Structures using a General Higher-Order Plate and One-Dimensional Theories for Classical and Cosserat Continuum Having Constrained Microrotation

Abstract

In this study nonlinear finite element models for beams and plates considering general higher-order expansions of the displacement fields have been developed, The models account for Cosserat continuum having constrained micro-rotation. The models can be used to analyze solid continua with very small inclusions or small scale structures in which material length scales, that classical continuum mechanics fails to capture, play a role. The beam and plate models developed herein are used to study the effect of different length scale parameters and the orientation of small inclusions. Also, the classical plate theory for rotation gradient dependent potential energy (Cosserat continuum for constrained micro-rotation) is applied to model nano-indentation on a carbon nanotube (CNT)-reinforced hard coating on an elastic substrate to see the effect of CNT reinforcement, which is modeled by small material length scale parameters. A general higher-order one-dimensional theory has also been developed in cylindrical and curvilinear cylindrical coordinate systems by considering a very general displacement approximation of arbitrary cross-section of a body in polar coordinates. Based on this approximation, the governing equations of motion have been derived using the principle of virtual displacements for large deformation case. Further, a nonlinear finite element model is developed to determine nonlinear response using the theories presented. In the numerical examples, the finite element model is used to analyze shell and rod-like structures for large deformation. Also, these higher-order one-dimensional theories are very relevant for the analysis of shell and rod-like structures of Cosserat continuum for constrained micro rotation because all gradient elasticity theories require C1 or higher-order continuity of the displacement variables, which is hard to achieve in the case of two or three dimensions, especially for non-rectangular grids. The one-dimensional theory developed herein allows continuity of any desired order of the variables by general Hermite interpolation functions in the finite element model.