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

2. 7. Bridging Scale Numerical Examples

3. 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

4. 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)

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

6. 7.2 1D Bridging Scale Numerical Examples – Lennard-Jones Initial MD & FEM displacements MD impedance force (applied correctly) MD impedance force (not applied correctly)

7. 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

8. 1D Wave Propagation - Energy Transfer 99.97% of total energy transferred from MD domain Only 9.4% of total energy transferred without impedance force

9. 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

10. 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

11. 2D Wave Propagation Snapshots of wave propagation

12. 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.

13. 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

14. 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  t FE =40  t MD Ramp velocity BC on FEM Time Velocity t1t1 V max

15. 2D Dynamic Crack Propagation Beginning of crack opening Crack propagation just before complete rupture of specimen

16. 2D Dynamic Crack Propagation Bridging scale potential energy Full MD potential energy

17. 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

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

19. 7.6 Dynamic Crack Propagation in Three Dimensions 3D FCC lattice Lennard Jones 6-12 potential Each FEM = 200 atoms 1000 FEM, 117000 atoms Fracture initially along (001) plane Time Velocity t1t1 V max FEM MD+FEM Pre-crack V(t)

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

21. MD/Bridging Scale Comparison Full MD Bridging Scale

22. MD/Bridging Scale Comparison Full MD Bridging Scale

23. MD/Bridging Scale Comparison Full MD Bridging Scale

24. MD/Bridging Scale Comparison Full MD Bridging Scale

25. 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

26. 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