E. B. Tadmor, M. Ortiz and R. Phillips
Philosophical Magazine A, 73, 1529–1563 (1996).
We develop a method which permits the analysis of problems requiring
the simultaneous resolution of continuum and atomistic length scales-and
associated deformation processes-in a unified manner. A finite element
methodology furnishes a continuum statement of the problem of interest and
provides the requisite multiple-scale analysis capability by adaptively
refining the mesh near lattice defects and other highly energetic regions.
The method differs from conventional finite element analyses in that
interatomic interactions are incorporated into the model through a crystal
calculation based on the local state of deformation. This procedure endows
the model with crucial properties, such as slip invariance, which enable
the emergence of dislocations and other lattice defects. We assess the
accuracy of the theory in the atomistic limit by way of three examples:
a stacking fault on the (111) plane, and edge dislocations residing on (111)
and (100) planes of an aluminium single crystal. The method correctly predicts
the splitting of the (111) edge dislocation into Shockley partials. The
computed separation of these partials is consistent with results obtained by
direct atomistic simulations. The method predicts no splitting of the Al Lomer
dislocation, in keeping with observation and the results of direct atomistic
simulation. In both cases, the core structures are found to be in good
agreement with direct lattice statics calculations, which attests to the
accuracy of the method at the atomistic scale.