## Abstract

We describe a method of constructing exact solutions of the equations of molecular dynamics in non-equilibrium settings. These solutions correspond to some viscometric flows, and to certain analogs of viscometric flows for fibers and membranes that have one or more dimensions of atomic scale. This work generalizes the method of objective molecular dynamics (OMD) (Dumitricǎ and James, 2007). It allows us to calculate viscometric properties from a molecular-level simulation in the absence of a constitutive equation, and to relate viscometric properties directly to molecular properties. The form of the solutions is partly independent of the form of the force laws between atoms, and therefore these solutions have implications for coarse-grained theories. We show that there is an exact reduction of the Boltzmann equation corresponding to one family of OMD solutions. This reduction includes most known exact solutions of the equations of the moments for special kinds of molecules and gives the form of the molecular density function corresponding to such flows. This and other consequences leads us to propose an addition to the principle of material frame indifference, a cornerstone of nonlinear continuum mechanics. The method is applied to the failure of carbon nanotubes at an imposed strain rate, using the Tersoff potential for carbon. A large set of simulations with various strain rates, initial conditions and two choices of fundamental domain (unit cell) give the following unexpected results: Stone-Wales defects play no role in the failure (though Stone-Wales partials are sometimes seen just prior to failure), a variety of failure mechanisms is observed, and most simulations give a strain at failure of 15-20%, except those done with initial temperature above about 1200 K and at the lower strain rates. The latter have a strain at failure of 1-2%.

Original language | English (US) |
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Pages (from-to) | 145-163 |

Number of pages | 19 |

Journal | Journal of the Mechanics and Physics of Solids |

Volume | 58 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2010 |

### Bibliographical note

Funding Information:We acknowledge valuable discussions with Stuart Antman, Traian Dumitrică, and Stefan Müller. The work of Kaushik Dayal was supported by AFOSR (FA9550-09-1-0393) and a Minnesota Supercomputing Institute Research Scholarship. The work of R.D.J. was supported by AFOSR (FA9550-09-1-0339 and GameChanger FA9550-07-1-0462), DoE (DE-FG02-05ER25706) and NSF (DMS-0757355). This research was also supported in part by the National Science Foundation through TeraGrid resources provided by Pittsburgh Supercomputing Center.

## Keywords

- Carbon nanotubes
- Dynamic fracture
- Molecular dynamics
- Objective structures
- Rheology