Non-linear finite element implementation of micropolar hypo-elastic materials

Abstract

In this paper, updated Lagrangian finite element formulations for large elastic deformation of micropolar hypo-elastic materials are presented. Using representation theorems of tensor functions, the general form of the constitutive equations for the micropolar hypo-elastic materials model are presented. The finite element formulations are based on the general form of the micropolar hypo-elastic constitutive equations in conjunction with Jaumann rate and a new rate called gyration rate. Gyration rate describes the deformation of the material in view of an observer attached to the micro-structure. An incrementally objective stress and couple stress update procedure is developed and its applicability is investigated based on the Jaumann and gyration rates. Two planar examples, based on the simplified forms of the proposed constitutive equations with Jaumann and gyration rates are investigated. In the first example, a plate with a small circular hole is analyzed and it is shown that the results of non-linear analysis converge to the linear micropolar elasticity for the case of infinitesimal deformations. In another example, 2D deformation of a cantilever beam which is modeled as a slender rectangle is investigated. The results of Jaumann and gyration rates are very close to each other, but an increase in the micropolar material parameters, particularly an increase in the length-scale parameter results in the reduction of the deformation of the beam. Also, it is shown that for a specimen with very small dimensions, e.g. in a micro-beam, the effect of the length-scale parameter is more sensible than that of a specimen with an ordinary or large dimensions. This means that micropolar effects become important for deformations taking place at small scales.