Mixed Atomistic and Continuum Models of Deformation in Solids†

The modeling of processes involving multiple length scales is an area of pressing concern, especially in problems such as nanoidentation and crack tip dislocation activity. In these cases, there is more than one characteristic dimension with the nanometer scale arising due to the presence of extended defects such as dislocations and a second length scale at least 2 orders of magnitude larger set by the scale of the indenter or the crack tip itself. To properly model such processes, both scales must be treated explicitly, which is normally beyond the scope of conventional atomistic and continuum analyses alike. This paper describes a quasicontinuum method which seizes upon the strengths of both atomistic and continuum techniques and allows for the simultaneous treatment of multiple scales. The method is based upon a continuum formulation of the problem of interest as a boundary value problem treated within the confines of the finite element method. We part company with traditional approaches by utilizing direct atomistic calculations as the source of the constitutive input used in the finite element analysis. The method is illustrated through application to the case of the structure and energetics of single dislocations. This case is a stringent test as it represents an extreme limit for the model since dislocation core structures are primarily dictated by lattice effects. It is then shown how the method may be applied to problems of tribological concern such as nanoindentation, where it is found that dislocations are initiated beneath the indenter.