Computational materials science plays a crucial role in advancing new and improved materials. To leverage the advantages of local and nonlocal methods and aid in the advancement of predictive capabilities for materials, multiscale models have been introduced. Many such methods have been proposed to overcome computational challenges in accuracy and efficiency. In this work, I begin by presenting a review of some multiscale methods for crystalline modeling to provide context for this dissertation.
Together with my advisor Dr. Xingjie Helen Li, we explore the static behavior of a bottom-up nonlocal-to-local coupling method, Atomistic-to-Continuum coupling, and explore the dynamic behavior of a nonlocal method, Peridynamics, to explore a bimaterial interface.
Inspired by the blending method developed by \cite{Seleson2013} for nonlocal-to-local coupling, we create a symmetric and consistent blended force-based Atomistic-to-Continuum (AtC) scheme for one-dimensional atomistic chains. AtC coupling schemes have been introduced to utilize the accuracy of atomistic models near known defects and the computational efficiency of continuum models elsewhere. The conditions for the well-posedness of the underlying model are established by analyzing an optimal blending size and blending type to ensure the stability of the $H^1$ seminorm for the blended force-based operator. We present several numerical experiments to test and confirm the theoretical findings.
Then, we create a Peridynamics-to-Peridynamics scheme to model a bimaterial bar in one dimension. Peridynamics (PD) naturally allows for the simulation of crack propagation in its model due to its use of integro-differentials and time derivatives instead of the spatial derivatives typical of classical models. Although PD can be computationally intensive, its ability to accurately model fracture behavior, especially at material interfaces, makes it a valuable tool for achieving high accuracy in simulations, especially due to the susceptibility of fracture where differing materials meet. We prove the conservation laws, derive the dispersion relation, and estimate the coefficient of reflection near the interface for this nonlocal-to-nonlocal problem. We seek an optimal nonlocal interaction kernel in the governing equation for the cross-material interaction to reduce spurious artifacts when the kernel is assumed to be constant.
Lastly, I discuss potential future development in Atomistic-to-Continuum coupling and Peridynamics.