An implicit Material Point Method for micropolar solids undergoing large deformations

Abstract

Modelling the mechanical behaviour of structural systems where the system size approaches that of the material microstructure (such as in MEMS) presents challenges to the standard continuum assumption and classical models can fail to predict important phenomena. Of the various non-conventional continuum frameworks developed to tackle this issue, the micropolar (Cosserat) continuum is widely acknowledged as a suitable and rigorous alternative for its ability to naturally predict size effects by introducing characteristic length scales. This work proposes an implementation of geometrically non-linear micropolar theory using an implicit Material Point method, for the purpose of simulating nanoscale large-deformation problems involving Hookean materials. The framework employs an analytically-derived consistent tangent, and is verified with a novel benchmark problem derived using the Method of Manufactured Solutions. Due to similarities between the methods, many aspects of the formulation could be used to construct an Updated Lagrangian Finite Element Method.