• Interaction and multiscale mechanics
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• v.7 no.1
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• pp.543-561
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• 2014

We present a full energy and force formulation of the quasicontinuum method with non-local and local transition elements. Non-local transition elements are developed to transmit inhomogeneity from the atomistic to the continuum regions. Local transition elements are developed to resolve the mathematical mismatch between non-local atoms and the local continuum. The rationale behind these transition elements is provided by analyzing the energy and force transitions between atoms and continuum under the Cauchy-Born rule. We show that breakdown of the Cauchy-Born rule occurs for slaved atoms of local elements within the cutoff of non-local atoms. The inadequacy of the Cauchy-Born rule at the transition region naturally leads to the need of atomistic treatment of transition slaved and transition representative atoms. Such an atomistic treatment together with a full or cutoff sampling allows non-local transition elements containing these transition entities to transmit inhomogeneity. Different force formulations for transition representative atoms and pure local representative atoms allow the local transition elements to resolve non-local and local mismatches. The method presented herein is validated by force calculations in an unstressed perfect crystal as well as an unrelaxed grain boundary model. A nanoindentation simulation in 3D is conducted to demonstrate the accuracy and efficiency of the proposed method.

• Proceedings of the KSME Conference
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• 2004.04a
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• pp.542-546
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• 2004

Fully non-local Quasicontinuum method using sub-divided region with Hermite interpolation function is proposed for simulation of carbon nanotube. Tersoff-Brenner potential is adopted for interaction of bonded atoms and also van der Waals force for non-bonded interaction. Bending of single wall carbon nanotube with chirality (20,0) and 15nm length is simulated up to 90 degree. Bending of double wall carbon nanotube with chirality (20,0) and (12,0) is simulated up to 65 degree. Bending of four wall carbon nanotube is simulated up to 45 degree.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2007.04a
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• pp.415-420
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• 2007

In analyzing the nano-scale behavior of nano devices or materials, QC method is efficient because it does not treat all the atoms. But for more accurate analysis in QC method, it is important to consider temperature and surface effects. In finite temperature, free energy is considered instead of potential energy. Because the surface area to volume ratio increases as the length scale of a body decreases, the surface effects are more dominant. In this paper, temperature related Cauchy-Born rule and surface Cauchy-Born rule are proposed to configurate the strain energy density. This method is applied to small and homogeneous deformation in two dimensional problem using finite element simulation.

• Proceedings of the KSME Conference
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• 2004.11a
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• pp.400-403
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• 2004

We present a coarse-graining model to describe the mechanical behaviors of multi-walled carbon nanotubes. To find the atomic configuration in membrane-like nanostructure i.e. carbon nanotube, we employ interpolation functions and the associated element-variables that are defined in the subdivided region. Tersoff-Brenner potential is adopted for interaction of bonded atoms and also van der Waals force for non-bonded interaction. Moreover, we simulate the coarse-graining multi-walled carbon nanotubes with defects and its result is compared with that of perfect multi-walled carbon nanotubes.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2006.04a
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• pp.395-402
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• 2006

In analyzing the nano-scale phenomena or behaviors of nano devices or materials, it is often desirable to deal with more atoms than can be treated only with a full atomistic simulation. However, even now, it is advisable to apply the atomistic simulation to the narrow region where the deformation field changes rapidly but to apply the conventional continuum model to the region far from that region. This equivalent continuum model can be formulated by applying the Cauchy-Born rule to the exact atomistic potential as in the quasicontinuum method. To couple the atomistic model with the equivalent continuum model, continuum displacements are conformed to the molecular displacements at the discrete positions of the atoms within the bridging domain. To satisfy the coupling constraints, we apply the Lagrange multiplier method. The continuum model in the bridging model should be applied on the region where the deformation field changes gradually. Then we can make the nodal spacing in the continuum model be much larger than the atomic spacing. In the first step, we generate the atomic-resolution mesh with the nodal spacing equal to the atomic spacing, and then we eliminate the nodal degrees of freedom adaptively using the node deactivation techniques. We eliminate more DOFs as the regions are more far from the atomistic region. Computing time and computational resources can be greatly reduced by the present node deactivation technique in multi scale analysis.