Multiresolution molecular mechanics: Surface effects in nanoscale materials

Abstract

Surface effects have been observed to contribute significantly to the mechanical response of nanoscale structures. The newly proposed energy-based coarse-grained atomistic method Multiresolution Molecular Mechanics (MMM) [Q. Yang, A.C. To, Comput. Methods in Appl. Mech. Eng. 283 (2015) 384-418] is applied to capture surface effect for nanosized structures by designing a surface summation rule SR S within the framework of MMM. Combined with previously proposed bulk summation rule SR B , the MMM summation rule SR MMM is completed. SR S and SR B are consistently formed within SR MMM for general finite element shape functions. Analogous to quadrature rules in finite element method (FEM), the key idea to the good performance of SR MMM lies in that the order or distribution of energy for coarse-grained atomistic model is mathematically derived such that the number, position and weight of quadrature-type (sampling) atoms can be determined. Mathematically, the derived energy distribution of surface area is different from that of bulk region. Physically, the difference is due to the fact that surface atoms lack neighboring bonding. As such, SR S and SR B are employed for surface and bulk domains, respectively. Two-and three-dimensional numerical examples using the respective 4-node bilinear quadrilateral, 8-node quadratic quadrilateral and 8-node hexahedral meshes are employed to verify and validate the proposed approach. It is shown that MMM with SR MMM accurately captures corner, edge and surface effects with less 0.3% degrees of freedom of the original atomistic system, compared against full atomistic simulation. The effectiveness of SR MMM with respect to high order element is also demonstrated by employing the 8-node quadratic quadrilateral to solve a beam bending problem considering surface effect. In addition, the introduced sampling error with SR MMM that is analogous to numerical integration error with quadrature rule in FEM is very small.