Mixed Atomistic and Continuum Models of Deformation in Solids
E. B. Tadmor, R. Phillips and M. Ortiz
Langmuir, 12, 4529–4534 (1996).


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.