Browsing by Subject "Beltrami representation"

Now showing 1 - 1 of 1
  • Thumbnail Image
    Item
    Results on the interaction between atomistic and continuum models
    (2014-08) Admal, Nikhil Chandra
    In this thesis, we develop continuum notions for atomistic systems which play an important role in developing accurate constitutive relations for continuum models, and robust multiscale methods for studying systems with multiple length and time scales. We use a unified framework to study the Irving--Kirkwood and Murdoch--Hardy procedures used to obtain definitions for continuum fields in atomistic systems. We identify and investigate the following three problems. 1. Continuum fields derived for atomistic systems using the Irving--Kirkwood or the Murdoch--Hardy procedures correspond to a spatial description. Due to the absence of a deformation mapping field in atomistic simulations, it is uncommon to define atomistic fields in the reference configuration. We show that the Murdoch--Hardy procedure can be modified to obtain pointwise continuum fields in the reference configuration using the motion of particles as a surrogate for the deformation mapping. In particular, we obtain definitions for the first and second atomistic Piola--Kirchhoff stress tensors. An interesting feature of the atomistic first Piola--Kirchhoff stress tensor is the absence of a kinetic contribution, which in the atomistic Cauchy stress tensor accounts for thermal fluctuations. We show that this effect is also included in the atomistic first Piola--Kirchhoff stress tensor through the motion of the particles. 2. We investigate the non-uniqueness of the atomistic stress tensor stemming from the non-uniqueness of the potential energy representation. In particular, we show using rigidity theory that the distribution associated with the potential part of the atomistic stress tensor can be decomposed into an irrotational part that is independent of the potential energy representation, and a traction-free solenoidal part. Therefore, we have identified for the atomistic stress tensor a discrete analog of the continuum generalized Beltrami representation (a version of the vector Helmholtz decomposition for symmetric tensors). 3. We show that an ambiguity in the original Irving--Kirkwood procedure resulting due to the non-uniqueness of the energy decomposition between particles can be completely avoided through an alternate derivation for the energy balance. It is found that the expressions for the specific internal energy and the heat flux obtained through the alternate derivation are quite different from the original Irving--Kirkwood procedure and appear to be more physically reasonable. Next, we apply spatial averaging to the pointwise field to obtain the corresponding macroscopic quantities. These lead to expressions suitable for computation in molecular dynamics simulations.