Chemistry Reference
In-Depth Information
tungsten), both on cleavage and non-cleavage planes. Failure mechanisms for
brittle fracture in nickel have been investigated as well, under mixed-mode
loading conditions.
82-86
The Quasi-continuum Method
The quasi-continuum (QC) method was first introduced in 1996 by
Tadmor et al.
87,88
for the investigation of deformation in solids. Ever since,
this method has been one of the most powerful and widely applied hybrid meth-
odologies. Its primary applications include the study of dislocation nucleation,
cracks, interfaces, grain boundary structure and deformation, nanoindentation
phenomena, and so on. Various applications are discussed in more detail below.
Since its appearance, the model has been improved and expanded,
89-94
and
these more complete versions are briefly presented here. If additional details
are needed, several specialized reviews are available.
71,95-98
The idea behind the QC method is to consider the atomistic description
(classical or quantum mechanically based) as being the correct way to model a
material. Continuum assumptions are then progressively introduced to reduce
the degrees of freedom of the system, making it computationally tractable.
However, a fully atomistic resolution can be maintained wherever it is needed.
The reduction of degrees of freedom occurs only through the introduction of
kinematic constraints, i.e., where most of the atomic displacements are com-
pletely determined by the calculated positions of a much smaller subset of
atoms. As a consequence, and this is one of the key features of the model,
the coupling between the continuum and the atomistic regions is almost or
completely seamless (depending upon which implementation of the method
is considered). The coupling is seamless because the only difference between
the two regions is the degree of coarsening, i.e., the constitutive properties
of the continuum, as well as those of the atomic region, are always obtained
from the atomistic level. This feature also garantees that key properties of the
material, such as crystal symmetries and slip invariance, are automatically
included. This makes the QC treatment of the continuum region significantly
different
from traditional
continuum calculations, where
constitutive
properties are derived from assumed phenomenological forms.
A second key feature of the method is adaptive meshing,
99
i.e., the ability
to shift the location of the atomistic/continuum boundary and to refine or
coarsen the continuum grid during a single simulation. This feature makes
the QC method particularly suited for investigating phenomena such as the
diffusion of defects.
Lastly, any underlying atomistic model can be used, whether quantum
mechanically or classically based. In practice, semi empirical interatomic
potentials such as EAM
9,10
and Stillinger-Weber
8
(three-body interaction)
potentials have usually been used to model the atomistic regime.
The reduction of the degrees of freedom in the QC method is based on
the assumption that, under a smoothly changing deformation gradient, it is not
