A definition of the atomic level stress is important both for physical interpretation of continuum fields and understanding of molecular scale simulations.  In order to establish atomic to continuum linkage, it is also necessary to define conjugate local strains at the atomic level.  Particular examples highlighting the importance of these concepts include an understanding of local structure and mechanical behavior for materials without long range order, and reconciliation of atomic level residual stress with experimentally measured lattice mismatch strain.  Current bridging methods only employ lattice mismatch by comparing mean positions of atoms between a reference and current state in order to define local atomic level strains.  We introduce here a local strain measure called vibrational strain defined by comparing the shape of the atomic cloud traced by the thermal vibration of a particular atom between current and reference configurations.  The vibrational strain captures the kinematics of lattice defects and structural heterogeneity as well as thermal effects, both of which are ignored by lattice mismatch strain calculated using mean atomic positions.  The choice of the thermal vibration to define such a strain is motivated by the following observations from molecular dynamics simulation in the NPT ensemble on a periodic fcc Aluminum crystal with a vacancy in the center of the supercell: (a) only a negligible percentage of the energy associated with atomic level residual stress is attributable to the lattice mismatch strain, (b) the direction of the primary eigenvector direction for the vibrational strain matches that of the residual stress, (c) the spatial variation of the vibrational strain closely parallels that of the local potential energy per atom.  The supplemental strain defined in this work is necessary to establish energy equivalence between the discrete and continuous descriptions, so that continuum scale concepts may be meaningfully applied at small length scales.

Atomic level strain to account for structural heterogeneity and thermal effects

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