Chemistry Reference
In-Depth Information
Sizewise, for three-dimensional problems zone II is actually a surface
(a line for a 2D problem), while zone III should be sufficiently wide to provide
a complete set of neighbors to those atoms in III that interact with atoms in II.
This means that the width of zone III should be at least equal to the cutoff
distance of the interatomic potential that is being used in the calculations.
However, for density-dependent potentials or potentials containing three-
body terms, the zone III width should be twice that distance.
Lastly, as pointed out earlier, no matching of the forces between the
atomistic and the continuum regions has been introduced. Therefore, it is
necessary to impose additional conditions to guarantee stress equilibrium
between the two domains. To do so, the elastic energy is first expanded in a
Taylor series with respect to the strain
E
:
0
E
ij
þ
0
E
ij
E
kl
2
E
qE
ij
E
kl
Þþ
q
E
qE
ij
1
2
q
E
ð
E
Þ¼
E
ð
0
½
1
0
E
ij
E
kl
E
mn
þ ...
3
E
qE
ij
E
kl
E
mn
1
6
q
þ
then all coefficients are set equal in the atomistic and continuum regions (the
strains had already been set equal because of the coupling procedure). This
condition is therefore equivalent to the matching of the elastic constants
between the two media:
0
¼
q
E
qE
c
ij
first order
ij
0
2
E
qE
ij
E
kl
c
ijkl
¼
q
second order
½
2
0
3
E
qE
ij
E
kl
E
mn
q
c
ijklmn
¼
third order
...
The main approximation of the FEAt coupling method is therefore given
by the truncation of the Taylor series expansion [1]. Within the framework of
linear local elasticity theory, it is enough to retain (and match) terms only up
to the second order. However, if nonlinear elastic effects need to be included,
third-order elastic constants must also be matched. Moreover, because the
first-order elastic constants in the continuum are zero by definition, such a
matching condition requires the interatomic potential to yield zero stress in
a perfect lattice.
Applications
The FEAt methodology has been extensively applied to the
study of crack propagation in body-centered cubic (bcc) metals (iron and
