Hierarchial Models of Nanomechanics and Micromechanics

An overview of theoretical and computational methods in the rapidly advancing field of Nanomechanics and Micromechanics of materials is given. This overview emphasizes the Multiscale Modelling of Materials (MMM) approach, which relies on a reduction of the degrees of freedom at natural length scales. Two main challenges are identified in the MMM approach: (i) the computational complexity of fully-coupled simulations, and (ii) inherent difficulties in describing the temporal evolution of nano and microsystems with vastly different time-scales. We first introduce the fundamental methods of quantum mechanics for description of the electronic degrees of freedom. Essential concepts are first introduced, followed by a presentation of the Local Density Approximation (LDA) and Density Function Theory (DFT), pointing out to both limitations and successful applications. Connections between ab intio methods and approximate interatomic potentials used in Molecular Dynamics (MD) simulations are discussed. Atomistic simulation methods ( e.g. MD, Langevin Dynamics (LD) and the Kinetic Monte Carlo (KMC)) and their applications at the nanoscale are then discussed. When the length scale is too large for direct atomistic simulations and too small to allow homogenization by continuum methods, reduction of the degrees of freedom is accomplished by simulations of defect interactions, thus eliminating all electronic and atomic degrees of freedom in this meso-scale. The role played by Dislocation Dynamics (DD) and Statistical Mechanics (SM) methods in understanding microstructure self-organization, heterogeneous plastic deformation, material instabilities and failure phenomena is also presented. Applications of the methods presented here are given for a number of nano and micro material systems.