doi:10.1016/j.cma.2003.12.008 
Copyright © 2004 Elsevier B.V. All rights reserved.
An introduction to computational nanomechanics and materials
W. K. Liu
, 
, 1, E. G. Karpov, S. Zhang and H. S. Park
Department of Mechanical Engineering, Northwestern University, The Technological Institute, 2145 Sheridan Road, Evanston, IL 60208-3111, USA
Received 1 June 2003;
Revised 15 October 2003;
accepted 2 December 2003.
Available online 5 February 2004.
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Abstract
Many arenas of research are rapidly advancing due to a combined effort between engineering and science. In some cases, fields of research that were stagnating under the exclusive domain of one discipline have been imbued with new discoveries through collaboration with practitioners from the second discipline. In computational mechanics, we are particularly concerned about the technological engineering interest by combining engineering technology and basic sciences through modeling and simulations. These goals have become particularly relevant due to the emergence of the field of nanotechnology, and the related burst of interest in nanoscale research. In this introductory article, we first briefly review the essential tools used by nanoscale researchers. These simulation methods include the broad areas of quantum mechanics, molecular dynamics and multiple-scale approaches, based on coupling the atomistic and continuum models. Upon completing this review, we shall conclusively demonstrate that the atomistic simulation tools themselves are not sufficient for many of the interesting and fundamental problems that arise in computational mechanics, and that these deficiencies lead to the thrust of multiple-scale methods. We summarize the strengths and limitations of currently available multiple-scale techniques, where the emphasis is made on the latest perspective approaches, such as the bridging scale method, multi-scale boundary conditions, and multi-scale fluidics. Example problems, in which multiple-scale simulation methods yield equivalent results to full atomistic simulations at fractions of the computational cost, are shown. We conclude by discussing future research directions and needs in multiple-scale analysis, and also discuss the ramifications of the integration of current nanoscale research into education.
Author Keywords: Nanomechanics and materials; Multi-scale simulations
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Fig. 1. Coordination in atomic systems.
Fig. 2. Pair-wise potentials and the interatomic forces: (a) Lennard–Jones, (b) Morse.
Fig. 3. Coordination in the hydrogen molecule.
Fig. 4. Coordination in the hydrogen ion.
Fig. 5. Ion-beam deposition: (a) waves due to ion–lattice collisions are reflected from the boundaries; (b) the collisions heat up the lattice causing unrealistic bouncing of the deposited ions.
Fig. 6. Indentation pattern in a golden substrate: MD simulation. Actual imprint size can be tens-to-hundreds of nanometers.
Fig. 7. A potential energy profile for MD fracture simulation: the fracture dynamics is affected by elastic waves, emitted by the crack tip (left) and reflected back by the MD domain boundaries (right).
Fig. 8. Friction behavior of a hydrocarbon system [109]: (a) simulation model of friction between two molecular surfaces; (b) comparison of the friction coefficients for hydrogen-terminated carbon surfaces and clean carbon surfaces.
Fig. 9. Using varying time-step in the simulation of ion-beam deposition. (a) Maximum velocity of atoms in the system. The peaks correspond to the deposition of an energetic ion. (b) Time step in the simulation based on the maximum velocity profile.
Fig. 10. Hierarchical modeling of Cybersteel [126]: (a) Quantum mechanics calculations yield the traction-separation law. (b) Concurrent modeling of the submicro-cell with embedded traction-separation law. (c) Concurrent modeling of the micro-cell with embedded constitutive law of the submicro-cell. (d) Modeling the fracture of the Cybersteel with embedded constitutive law of the micro-cell. (e) Fracture toughness and yield strength of the Cybersteel as a function of decohesion energy, determined by geometry of the nanostructures. (f) Snap-shots of the localization induced debonding process. (g) Experimental observations.
Fig. 11. Multi-scale analysis of a 15-walled CNT by a bridging scale method: (a) The multi-scale simulation model consists of 10 rings of carbon atoms (with 49,400 atoms each) and a meshfree continuum approximation of the 15-walled CNT by 27,450 nodes. (b) The global buckling pattern captured by meshfree method whereas the detailed local buckling of the ten rings of atoms are captured by a concurrent bridging scale molecular dynamic simulation.
Fig. 12. Illustration of the bridging scale approach: the MD and FE solutions are coupled through the projection technique. The ubiquitous atomistic resolution is replaced with a reduced MD region by utilizing the impedance boundary conditions. The dashed red line shows boundary of the domain of interest.
Fig. 13. Wave propagation through the atomistic domain in the FCC lattice structure: (a) impedance boundary conditions are involved at the MD/continuum interface, (b) continuity boundary conditions.
Fig. 14. Bridging multi-scale modeling of crack propagation: (a) initiation of a crack in the MD region, (b) pre-separation phase, (c) lattice dislocation pattern at the crack tip.
Fig. 15. Comparison of crack simulations at the lattice separation stage: (a) the full MD model, (b) the bridging multi-scale model. (c) The subdomain of coexisting of the MD and FE solutions (zoom in).
Fig. 16. Bridging multi-scale simulation of shear localization in a continuous bar.
Fig. 17. Snapshot of temperature gradient in a 1D thermal wave propagation process; comparison made for the full atomistic resolution (red) and the bridging multi-scale model (blue).
Fig. 18. An illustration to the concept of multi-scale boundary conditions: behavior of the MD boundary is governed by a deformable boundary equation, which accounts for the effect of a coarse scale domain.
Fig. 19. Multi-scale boundary conditions for nanoindentation problems.
Fig. 20. Plane MD/coarse scale interface in a 2D cubic lattice. Index m shows atomic numbering along the interface.
Fig. 21. Impedance kernel function for the 2D cubic lattice.
Fig. 22. Typical performance of the impedance boundary condition: dependence of the reflection coefficient on method parameters.
Fig. 23. The RBCs flow at 10 μm/s. at t=0 s, t=2.0 s, t=4.0 s, and t=6.0 s. The different colors indicate the value of stress exerted on RBCs. The aggregates of RBCs are clearly seen behind the narrow vessel.
Fig. 24. The shear of 4 RBCs at the shear rate of 3.0 s−1, at t=0 s, t=1.0 s, t=2.0 s, and t=3.0 s. The arrows show the fluid vortexes induced by the fluid-cell and cell–cell interactions.
Fig. 25. Three-dimensional simulation of a single red blood cell (essentially a hollow sphere for simplicity) squeezing through a capillary vessel.
Fig. 26. The history of the driven pressure during the squeezing process.
Table 1. Summary of the multiple-scale formulations
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