Atomistic simulation is a useful method for studying material science phenomena. Examination of the state of a simulated material and the determination of its mechanical properties is accomplished by inspecting the stress field within the material. However, stress is inherently a continuum concept and has been proven difficult to define in a physically reasonable manner at the atomic scale. In this paper, an expression for continuum mechanical stress in atomistic systems derived by Hardy is compared with the expression for atomic stress taken from the virial theorem. Hardy's stress expression is evaluated at a fixed spatial point and uses a localization function to dictate how nearby atoms contribute to the stress at that point; thereby performing a local spatial averaging. For systems subjected to deformation, finite temperature, or both, the Hardy description of stress as a function of increasing characteristic volume displays a quicker convergence to values expected from continuum theory than volume averages of the local virial stress. Results are presented on extending Hardy's spatial averaging technique to include temporal averaging for finite temperature systems. Finally, the behaviour of Hardy's expression near a free surface is examined, and is found to be consistent with the mechanical definition for stress.

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a b s t r a c tWe present an expression developed for calculating an atomic-scale deformation gradient within atomistic simulations. This expression is used to analyze the deformation fields for a one-dimensional atomic chain, a biaxially stretched thin film containing a surface ledge, and a FCC metal subject to indentation loading from a nanometer-scale indenter. The analyses presented show that the metric established here is consistent with the continuum mechanical concept of deformation gradient (which is known to have a zero curl for compatible deformations) in most instances. However, our metric does yield non-zero values of curl for atoms near loaded geometric inhomogeneities, such as those that form the ledges themselves and those beneath or adjacent to the indentation contact region. Also, we present expressions for higher order gradients of the deformation field and discuss the requirements for their calculation. These expressions are necessary for linking atomistic simulation results with advanced continuum mechanics theories such as strain gradient plasticity, thereby enabling fundamental, atomic-scale information to contribute to the formulation and parameterization of such theories.

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