Atomic to Continuum Passage for Nanotubes: A Discrete Saint-Venant Principle and Error Estimates
| Autor: | D. El Kass, Régis Monneau |
|---|---|
| Rok vydání: | 2014 |
| Předmět: |
Inverse function theorem
Saint venant Nanotube Mechanical Engineering Mathematical analysis Complex system Bending Carbon nanotube Condensed Matter::Mesoscopic Systems and Quantum Hall Effect law.invention Condensed Matter::Materials Science Mathematics (miscellaneous) Classical mechanics law Atom Continuum (set theory) Analysis Mathematics |
| Zdroj: | Archive for Rational Mechanics and Analysis. 213:25-128 |
| ISSN: | 1432-0673 0003-9527 |
| DOI: | 10.1007/s00205-014-0745-x |
| Popis: | We consider general infinite nanotubes of atoms in \({\mathbb{R}^3}\) where each atom interacts with all the others through a two-body potential. At the equilibrium, the positions of the atoms satisfy a Euler–Lagrange equation. When there are no exterior forces and for a suitable geometry, a particular family of nanotubes is the set of perfect nanotubes at the equilibrium. When exterior forces are applied on the nanotube, we compare the nanotube to nanotubes of the previous family. In part I of the paper, this quantitative comparison is formulated in our first main result as a discrete Saint-Venant principle. As a corollary, we also give a Liouville classification result. Our Saint-Venant principle can be derived for a large class of potentials (including the Lennard-Jones potential), when the perfect nanotubes at the equilibrium are stable. The approach is designed to be applicable to nanotubes that can have general shapes like, for instance, carbon nanotubes or DNA, under the oversimplified assumption that all the atoms are identical. In part II of the paper, we derive from our Saint-Venant principle a macroscopic mechanical model for general nanotubes. We prove that every solution is well approximated by the solution of a continuum model involving stretching and twisting, but no bending. We establish error estimates between the discrete and the continuous solution. More precisely we give two error estimates: one at the microscopic level and one at the macroscopic level. |
| Databáze: | OpenAIRE |
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