Topics in atomistic and continuum modeling and simulations of solids

This thesis consists of fours topics related to multiscale modeling and simulation of atomistic and continuum models for crystalline solids.; In the first part, we present an elementary and systematic discussion on the derivation of continuum theories from atomistic models for studying the elastic deformation of sheets, plates, and rods. The derivation is based on various generalizations of the classical Cauchy-Born rule. In particular, we discuss a so-called local Cauchy-Born rule which is very general and easy to use. As an application, we use the derived continuum models to study the elastic deformation of carbon nanotubes.; In the second part, accuracy of the quasicontinuum method is studied by reformulating the summation rules in terms of reconstruction schemes for the local environment of the representative atoms. The necessary and sufficient condition for uniform first order accuracy and eliminating the "ghost force" is formulated in terms of the reconstruction schemes. The quasi-nonlocal approach is discussed as a special case. The transition between atom-based and element-based summation rules is studied.; In the third part, we present some comparative studies on various boundary conditions for molecular dynamics simulations of crystalline solids. We focus on the effectiveness of these conditions in suppressing boundary reflections. As a quantitative comparison, we compute the reflection coefficients for these methods. Applications to fracture simulations are then demonstrated for problems including brittle cracks in a b.c.c. crystal and ductile cracks in a f.c.c. metal.; In the fourth part, we study brittle crack propagation using the framework of the heterogeneous multiscale method. Both the atomistic and the continuum models are formulated in the form of conservation laws. Whenever necessary the macroscale fluxes are computed using the microscopic model, which is in turn constrained by the local macro-state of the system through boundary conditions. Defect tracking and atomistic-based constitutive modeling strategies are developed.