Multiscale Methods for Crystalline Nanomaterials

University Of North Carolina At Charlotte, Charlotte NC

Investigators

Abstract

Crystal defects such as grain boundaries, cracks, or dislocations play significant roles in determining material properties and behaviors. To efficiently and reliably predict failure phenomena, fundamental level of descriptions at nanoscales are widely used to study how defects affect macroscopic properties such as elasticity and plasticity. Challenged by the computational limitations as well as high-fidelity requirements, this research project aims to develop and employ multiscale methods which can retain accuracy around defect cores while improving efficiency through local continuum descriptions. Two popular multiscale modeling strategies are: (1) bottom-up: coarse-graining of microscopic descriptions (e.g., atomistic models) of material behavior; (2) top-down: informing macroscopic models (e.g., continuum equations) with physics gleaned from the microscopic scales. The former provides "a closer" comparison with macroscopic experiments and the latter predicts the materials microscopic properties. This project focuses on several aspects of the multiscale modeling and mathematical analysis for both bottom-up and top-down approaches, aims at developing efficient and reliable coupling methods which smoothly integrate the microscopic and macroscopic descriptions, and establishing rigorous error estimates that will set precise guidelines for practical implementation (e.g., the optimal parameters used in the coupling mechanism and the finite element mesh used in the macroscopic scale). This project will also provide an opportunity to train both graduate and undergraduate students in the context of challenging and interdisciplinary research among applied mathematics, materials science and mechanical engineering. The overarching goal of this project is to develop multiscale methods for both bottom-up and top-down approaches with provable performance in terms of accuracy, efficiency, and reliability based on theoretical analysis and comprehensive error estimates. In particular, these are the four objectives: (1) The research will promote the understanding of bottom-up atomistic-to-continuum (AtC) methods for complex structured crystals. Successful outcomes of this project will greatly extend the applicability of existing AtC methods from simple lattice crystals to more complex ones. The comprehensive error analysis will lead to optimal coupling strategy and provide precise guidelines for practical implementations. (2) The research will also advance the developments of top-down nonlocal-to-local (NtL) continuum couplings. The new coupling framework is built based on geometric reconstruction, and will remove interfacial inconsistency while maintain all physical properties globally, such as conservation of energy and balance of linear momentum, in multi-dimensions, whereas none of existing coupling methods for NtL problems satisfies all of these properties. Furthermore, mathematical analysis will be built to ensure the well-posedness and reliability of modeling and computations. (3) This project will enhance the understandings and interconnections between the top-down and bottom-up approaches, and will also facilitate the exploitation of other scale-bridging algorithms in materials science. (4) High order and robust numerical schemes for large-scale problems will be developed to enhance the controls and designs of defects inside complex structured crystals, and improve the prediction of materials failures.