engleski

Finite element analysis of a linear elastic micropolar continuum: application of quadrilateral elements in 2D problems

In the attempt of finding an unified continuum theory various alternative continuum theories are developed. One of them is the micropolar (Cosserat) continuum theory. Unlike the classical (Cauchy) continuum theory where the interaction between two particles is described by means of the force vector field only, in the micropolar theory we presume the existence of an additional vector field which we call the moment stress vector as well as an independent microrotation field (which is completely independent of the macrorotation). With this assumption particles become orientable and we bring length scale into continuum theories. Due to the asymmetry of the stress and strain tensors, many new problems can be modelled, and the results obtained are much closer to the experimental results. In the framework of the finite element method, in this work we present the use of linked interpolation in the design of a micropolar continuum for elastic behaviour in two dimensions where the displacement interpolation is linked not only to element nodal displacements, but also to nodal microrotations. In the Timoshenko beam elements linked interpolation completely eliminates the shear-locking phenomenon and is capable of returning the exact solution for arbitrary polynomial loading. We generalise this family of interpolation functions for displacement fields that follow from the beam element analysis to membrane elements. We develop quadrilateral elements of different order where the displacement approximation depends not only on nodal displacement, but also on microrotations, while the microrotations are interpolated using conventional interpolation functions. In order to test stability and convergence of the presented finite elements, we check the element behaviour by performing patch tests of different order. It will be shown that modifications are needed in order to satisfy the criteria for monotonic element convergence.

micropolar continuum, microrotation, finite element method, linked interpolation, membrane elements

nije evidentirano