Simulation of cracks in a Cosserat medium using the extended finite element method

Abstract

This research investigates fracture behaviour in the Cosserat materials. The Cosserat elasticity description of the materials incorporates a characteristic length scale (e.g. grains, particles, fibres, etc.) into the model. The characteristic length scale in such materials is known to significantly influence the macroscopic behaviour of the whole body. Simulation of the fracture processes, such as crack opening and propagation, in Cosserat materials still remains a challenge for the scientific community. The goal of this thesis is to propose and validate a two dimensional extended finite element method model of edge cracks within the Cosserat elasticity theory framework.The crack modelling was conducted using the Finite Element Method (FEM) and eXtended Finite Element Method (XFEM) implemented in the Matlab code. The strong and weak formulations of the problem and the discrete XFEM equations are presented in the thesis. Mode I and II edge crack models in a Cosserat medium are discussed and verified through a series of patch and convergence tests. In addition, the numerical evaluation of the J-integral for the Cosserat medium is presented, and the J-integral for the Cosserat medium is compared to the J-integral for the classical elasticity.The XFEM/Cosserat method is shown to be robust and able to effectively model the edge crack problems in a Cosserat medium. Moreover, the elastic parameter α is found to be a powerful coupling tool between the microrotations and the translations. The Cosserat J-integral differs from the classical J-integral by 2% to 40%, for a given crack depending on the micropolar elastic coupling constant α.iii Acknowledgements I would like to thank my supervisors, Dr. Robert Gracie and Dr. Stanislav Potapenko, for their guidance and support during my study at the University of Waterloo.

Singularity-free defect mechanics for polar media

“…Several numerical methods have been used to study cracks within micropolar elasticity. Discrete element method, boundary element method [3,51], extended finite element method [24,59,60], and pseudospectral technique [53] are among those. Comparing the dislocation-and disclination-based approach and other methods, it is understood that once the analytical solution to the line defect is available, the dislocationand disclination-based approach seems to be more efficient and computationally less expensive.…”