Nano Mechanics and Materials: Theory, Multiscale Methods and Applications by Wing Kam Liu, Eduard G. Karpov, Harold S. Park

7. Bridging Scale Numerical Examples

7.1 Comments on Time History Kernel 1D harmonic lattice where indicates a second order Bessel function, is the spring stiffness, and the frequency, where m is the atomic mass  Spring stiffness utilizing LJ 6-12 potential where k is evaluated about the equilibrium lattice separation distance

Truncation Time History Kernel Impedance force due to salient feature of, an approximation can be made by setting the later components to zero Plots of time history kernel (full) Plots of time history kernel (truncated)

Comparison between time history integrals (full and truncated) Plot of time history kernel (full and truncated)

7.2 1D Bridging Scale Numerical Examples – Lennard-Jones Initial MD & FEM displacements MD impedance force (applied correctly) MD impedance force (not applied correctly)

1D Wave Propagation 111 atoms in bridging scale MD system 40 finite elements 10 atoms per finite element  t fe = 50  t md Lennard-Jones 6-12 potential Initial MD & FEM displacements

1D Wave Propagation - Energy Transfer 99.97% of total energy transferred from MD domain Only 9.4% of total energy transferred without impedance force

7.3 2D/3D Bridging Scale Numerical Examples Lennard-Jones (LJ) 6-12 potential Potential parameters  =  =1 Nearest neighbor interactions Hexagonal lattice structure; (111) plane of FCC lattice Impedance force calculated numerically for hexagonal lattice, LJ potential Hexagonal lattice with nearest neighbors

7.4 Two-Dimensional Wave Propagation MD region given initial displacements with both high and low frequencies similar to 1D example bridging scale atoms, full MD atoms 1920 finite elements (600 in coupled MD/FE region) 50 atoms per finite element Initial MD displacements

2D Wave Propagation Snapshots of wave propagation

2D Wave Propagation Final displacements in MD region if MD impedance force is applied. Final displacements in MD region if MD impedance force is not applied.

2D Wave Propagation Energy Transfer Rates: No BC: 35.47% n c = 0: 90.94% n c = 4: 95.27% Full MD: 100% n c = 0: 0 neighbors n c = 1: 3 neighbors n c = 2: 5 neighbors nn+1n+2n-1n-2

7.5 Dynamic Crack Propagation in Two Dimensions Problem Description: atoms, 1800 finite elements (900 in coupled region) Full MD = 180,000 atoms 100 atoms per finite element  t FE =40  t MD Ramp velocity BC on FEM Time Velocity t1t1 V max

2D Dynamic Crack Propagation Beginning of crack opening Crack propagation just before complete rupture of specimen

2D Dynamic Crack Propagation Bridging scale potential energy Full MD potential energy

2D Dynamic Crack Propagation Crack tip velocity/position comparison Full domain = 601 atoms Multiscale 1 = 301 atoms Multiscale 2 = 201 atoms Multiscale 3 = 101 atoms

Zoom in of Cracked Edge FEM deformation as a response to MD crack propagation

7.6 Dynamic Crack Propagation in Three Dimensions 3D FCC lattice Lennard Jones 6-12 potential Each FEM = 200 atoms 1000 FEM, atoms Fracture initially along (001) plane Time Velocity t1t1 V max FEM MD+FEM Pre-crack V(t)

Initial Configuration Velocity BC applied out of plane (z-direction) All non-equilibrium atoms shown

MD/Bridging Scale Comparison Full MD Bridging Scale

MD/Bridging Scale Comparison Full MD Bridging Scale

MD/Bridging Scale Comparison Full MD Bridging Scale

MD/Bridging Scale Comparison Full MD Bridging Scale

7.7 Virtual Atom Cluster Numerical Examples – Bending of CNT Global buckling pattern is capture by the meshfree method Local buckling captured by molecular dynamics simulation

VAC coupling with tight binding Comparison of the average twisting energy between VAC model and tight-binding model Meshfree discretization of a (9,0) single-walled carbon nanotube