Nano Mechanics and Materials: Theory, Multiscale Methods and Applications

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 (truncated) • Plots of time history kernel (full)

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 • MD impedance force (applied correctly) • Initial MD & FEM displacements • MD impedance force (not applied correctly)

1D Wave Propagation • 111 atoms in bridging scale MD system • 40 finite elements • 10 atoms per finite element • tfe = 50tmd • 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 • 30000 bridging scale atoms, 90000 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% • nc = 0: 90.94% • nc = 4: 95.27% • Full MD: 100% • nc = 0: 0 neighbors • nc = 1: 3 neighbors • nc = 2: 5 neighbors n-2 n-1 n n+1 n+2

Velocity Vmax t1 Time 7.5 Dynamic Crack Propagation in Two Dimensions • Problem Description: • 90000 atoms, 1800 finite elements (900 in coupled region) • Full MD = 180,000 atoms • 100 atoms per finite element • tFE=40tMD • Ramp velocity BC on FEM

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

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

FEM MD+FEM Velocity Vmax FEM t1 Time 7.6 Dynamic Crack Propagation in Three Dimensions V(t) • 3D FCC lattice • Lennard Jones 6-12 potential • Each FEM = 200 atoms • 1000 FEM, 117000 atoms • Fracture initially along (001) plane 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

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

Nano Mechanics and Materials: Theory, Multiscale Methods and Applications