Quasicontinuum Concurrent and Semi‐Analytical Hierarchical Multiscale Methods Across Atoms/Continuum

Abstract

In this chapter, quasicontinuum (QC) and semi-analytical multiscale methods for bridging atomistic and continuum scales are introduced. Various phenomena in nanoscale and their applications in nanotechnology are introduced. They include nanoindentation, dislocation initiation in crack tip and grain boundary, aluminum microtwinning, stress-induced phase transition, ferroelectric switching, development of atomistic-based continuum models with applications in hydrogen storage by nanocells, and mechanical, electrical and thermal properties of nanotubes. This chapter consists of three parts.Part 6.1: Using FEM as a link, basic concepts and methods in solid mechanics are introduced, such as the basic energy principle for materials and structures and the interpolation function to reduce degrees of freedom.Part 6.2: Introduction of the QC concurrent multiscale numerical method in terms of the atomistic-based energy density function derived from the Cauchy-Born rule and its various applications. Part 6.3: Development, evaluation and applications of atomistic-based continuum theory of carbon nanotubes through the hierarchical multiscale method.This chapter ends by using the Cauchy-Born rule to prove mathematically the inverse mapping rule of the GP method proposed in Section 5.5.