Elastic nanobeam modelled using peridynamics - length scale effects

The traditional multiscale approach couples two models operating at different scales. An alternative modelling strategy, called peridynamics originally developed by [1], is to continualize the molecular dynamic models, thus replacing inhomogeneities present on smaller length scales by an enhanced continuum description on larger length scales resulting in a nonlocal reformulation of continuum mechanics. Peridynamics is a single multiscale model valid over wide range of length scales and can be considered as an upscaling of molecular dynamics. Therefore peridynamics models should recover the same dynamics and preserve all characteristic properties of molecular dynamics, which are lost by classical continuum mechanics models. The advantage of the peridynamic models is that they can be solved more cheaply than the corresponding molecular dynamic models. Peridynamics is a generalized continuum theory employing a nonlocal model of force interaction. Each material point interacts with its neighborhood within a sphere, called the horizon that serves as an internal length scale in the model. The interaction between the material points is described by a bond force which is not an electrostatic force but can be related to the strain energy of classical continuum mechanics.
The objective of this study is to investigate whether the horizon in peridynamics can model the size dependence of the Young’s modulus at the nanoscale, shown experimentally and with molecular dynamic simulation in many previous works. Peridynamic simulations of copper beams with different sizes under traction have been performed employing the molecular dynamics code LAMMPS, see [2]. The force-displacement curves have been compared directly to molecular dynamics simulation and an appropriate value for the horizon has been chosen. The results showed that the Young’s modulus changes with the size and seems to tend to a limit which is the macroscopic value of the Young’s modulus for copper.