Material fields in atomistics as pull-backs of spatial distributions

The various fields defined in continuum mechanics have both a material and a spatial description that are related through the deformation mapping. In contrast, continuum fields defined for atomistic systems using the Irving-Kirkwood or Murdoch-Hardy procedures correspond to a spatial description. It is uncommon to define atomistic fields in the reference configuration due to the lack of a unique definition for the deformation mapping in atomistic systems. In this paper, we construct referential atomistic distributions as pull-backs of the spatial distributions obtained in the Murdoch-Hardy procedure with respect to a postulated deformation mapping that tracks particles. We then show that some of these referential distributions are independent of the choice of the deformation mapping and only depend on the reference and current configuration of particles. Therefore, the fields obtained from these distributions can be calculated without explicitly constructing a deformation map, and by construction they satisfy the balance equations. In particular, we obtain definitions for the first and second atomistic Piola-Kirchhoff stress tensors. We demonstrate the validity of these definitions through a numerical example involving finite deformation of a slab containing a notch under tension. An interesting feature of the atomistic first Piola-Kirchhoff stress tensor is the absence of a kinetic part, which in the atomistic Cauchy stress tensor accounts for thermal fluctuations. We show that this effect is implicitly included in the atomistic first Piola-Kirchhoff stress tensor through the motion of the particles. An open source program to compute the atomistic Cauchy and first Piola-Kirchhoff stress fields called MDStressLab is available online at http://mdstresslab.org.