A Multiscale
Approach
for
Modeling
Crystalline
Solids
A.
M.
Cuiti
̃
no
, L.
Stainier
, G.
Wang
, A.
Strachan
,
T. C ̧ a
̆
gın
, W. A.
Goddard,
III
, and
M.
Ortiz
Department
of
Mechanical
and
Aerospace
Engineering
Rutgers
Uni
versity
, Piscata
way, NJ
08854,
USA
Laboratoire
de
Techniques
A ́
eronautiques
et Spatiales
Uni
versity
of
Li`
ege,
4000
Li`
ege,
Belgium
Materials
and
Process
Simulation
Center
, Beckman
Institute
(139-74)
California
Institute
of
Technology
, Pasadena,
CA
91125,
USA
Graduate
Aeronautical
Laboratories
California
Institute
of
Technology
, Pasadena,
CA
91125,
USA
March
21,
2001
To appear
in:
Journal
of
Computer
Aided
Material
Design
Abstract
In
this
paper
we
present
a modeling
approach
to
bridge
the
atomistic
with
macroscopic
scales
in
crystalline
materials.
The
methodology
combines
identification
and
modeling
of
the
controlling
unit
processes
at
microscopic
level
with
the
direct
atomistic
determination
of
fundamental
material
proper
-
ties.
These
properties
are
computed
using
a man
y body
Force
Field
deri
ved
from
ab
initio
quantum-
mechanical
calculations.
This
approach
is exercised
to
describe
the
mechanical
response
of
high-purity
Tantalum
single
crystals,
including
the
effect
of
temperature
and
strain-rate
on
the
hardening
rate.
The
resulting
atomistically
informed
model
is found
to
capture
salient
features
of
the
beha
vior
of
these
crys-
tals
such
as:
the
dependence
of
the
initial
yield
point
on
temperature
and
strain
rate;
the
presence
of
a
mark
ed
stage
I of
easy
glide,
specially
at low temperatures
and
high
strain
rates;
the
sharp
onset
of
stage
II hardening
and
its
tendenc
y to
shift
towards
lower
strains,
and
eventually
disappear
, as
the
tempera-
ture
increases
or
the
strain
rate
decreases;
the
parabolic
stage
II hardening
at
low
strain
rates
or
high
temperatures;
the
stage
II softening
at high
strain
rates
or
low temperatures;
the
trend
towards
saturation
at
high
strains;
the
temperature
and
strain-rate
dependence
of
the
saturation
stress;
and
the
orientation
dependence
of
the
hardening
rate.
1
A.
M.
Cuiti
̃
no
et al.
A Multiscale
Approach
for
Modeling
Crystalline
Solids
1
Intr
oduction
This
paper
is
concerned
with
the
development
of
a multiscale
modeling
approach
for
adv
anced
materials
such
as
high-purity
bcc
single
crystals.
The
present
approach
is aligned
with
the
current
divide
and
conquer
paradigm
in
micromechanics
(see,
e.
g.,
[1,
2, 3, 4,
5,
6].
This
paradigm
first
identifies
and
models
the
controlling
unit
process
at
microscopic
scale.
Then,
the
ener
getics
and
dynamics
of
these
mechanisms
are
quantified
by
means
of
atomistic
modeling.
Finally
, the
macroscopic
dri
ving
force
is
correlated
to
macroscopic
response
via
microscopic
modeling.
This
last
step
involv
es
two stages,
localization
of
the
macroscopic
dri
ving
force
into
unit-process
dri
ving
forces
and
aver
aging
of
the
contrib
ution
of
each
unit
process
into
the
macroscopic
response.
We sho
w
in
this
article
that
the
meticulous
application
of
this
paradigm
renders
truly
predicti
ve mod-
els
of
the
mechanical
beha
vior
of
comple
x systems.
In
particular
we
predict
the
hardening
of
Ta single
crystal
and
its
dependenc
y for
a wide
range
of
temperatures
and
strain
rates.
The
feat
of
this
approach
is
that
predictions
from
these
atomistically
informed
models
reco
ver
most
of
the
macroscopic
characteristic
features
of
the
available
experimental
data,
without
a priori
kno
wledge
of
such
experimental
tests.
This
approach
then
pro
vides
a procedure
to
forecast
the
mechanical
beha
vior
of
material
in
extreme
conditions
where
experimental
data
is simply
not
available
or
very
dif
ficult
to
collect.
A
crucial
step
in
this
approach
is the
appropriate
selection
and
modeling
of
the
unit
processes.
These
models
supply
the
link
between
the
atomic
and
meso
scale
by
identifying
and
correlating
the
rele
vant
ma-
terial
properties,
susceptible
of
atomistic
determination
such
as
ener
gy
formation
for
defects,
with
the
cor
-
responding
dri
ving
forces.
In
this
case,
we
specifically
consider
the
follo
wing
unit
processes:
double-kink
formation
and
thermally
acti
vated
motion
of
kinks;
the
close-range
interactions
between
primary
and
forest
dislocation,
leading
to
the
formation
of
jogs;
the
percolation
motion
of
dislocations
through
a random
array
of
forest
dislocations
introducing
short-range
obstacles
of
dif
ferent
strengths;
dislocation
multiplication
due
to
breeding
by
double
cross-slip;
and
dislocation
pair
-annihilation.
A
set
of
material
parameters
is then
obtained
from
the
modeling
and
identification
stage,
which
is re-
quired
to
quantify
the
contrib
ution
of
each
of
the
unit
processes.
We compute
these
materials
properties
using
a combination
of
ab-initio
quantum
mechanics
(QM)
and
Force
Field
(FF)
calculations.
QM
describes
the
atomic
interactions
from
first
principles,
i.e.
with
no
input
from
experiments;
unfortunately
QM
methods
are
computationally
intensi
ve and
restricted
to
small
systems,
making
QM
calculations
impractical
to
study
most
of
the
materials
properties
governing
plasticity
. Force
Fields
give the
total
ener
gy
of
a system
as
a
potential
ener
gy
function
of
the
atomic
positions
and
with
Molecular
Dynamics
(MD)
allo
ws
the
simulation
of
systems
containing
millions
of
atoms.
We used
ab-initio
quantum
mechanical
calculations
(equations
of
state
of
various
crystalline
phases,
elastic
constants,
ener
getics
of
defects,
etc.)
to
develop
a man
y body
Force
Field
(FF)
(named
qEAM
FF)
for
Tantalum.
We use
the
qEAM
FF
with
MD
to
calculate
the
core
ener
gy
of
the
1/2a
111
scre
w dislocation,
that
of
the
edge
dislocation
with
Bur
gers
vector
b=1/2
a
111
in
(110)
planes.
We have also
calculated
the
formation
ener
gies
and
nucleation
lengths
of
the
kinks
in
b=1/2a
111
scre
w dislocations.
One
of
the
appealing
features
of
the
present
approach
is the
ability
to
incorporate
additional
unit
mech-
anisms
as
the
y may
be
required
by
the
physics
of
the
problem.
For
example,
formation
and
evolution
of
dislocation
structures
are
of
particular
interest
in
ductile
crystals
subjected
to
lar
ge
and
cyclic
deformation.
In
recent
studies
[7, 8], unit-mechanism
based
micromechanical
models
have been
proposed
to
elucidate
the
effecti
ve beha
vior
of
dislocation
structures
on
the
macroscopic
response.
The
organization
of
the
paper
follo
ws
the
sequential
stages
of
the
proposed
approach.
First,
we
pro
vide
a brief
description
of
each
of
the
unit
processes
including
the
governing
final
equations.
We then
identify
2
A.
M.
Cuiti
̃
no
et al.
A Multiscale
Approach
for
Modeling
Crystalline
Solids
l
p
τ
(a)
L
kink
(b)
Figure
1:
Schematic
of
the
double-kink
mechanism.
and
compute
by
atomistic
means
the
corresponding
material
properties.
Finally
, we
compare
the
predictions
against
experimental
data.
2
Unit
Pr
ocesses
Plastic
deformation
in
metallic
systems
is
the
macroscopic
manifestation
of
dislocation
acti
vity
. The
re-
sistance
to
the
dislocation
motion,
therefore,
engenders
the
hardening
properties
observ
ed
in
this
type
of
materials.
It is
then
the
comple
x interplay
of
microscopic
mechanisms
controlling
dislocation
mobility
,
dislocation
interaction
and
dislocation
evolution
which
confers
the
macroscopic
constituti
ve properties.
In
the
present
approach,
these
controlling
processes
are
considered
to
be
ortho
gonal
in
the
sense
that
are
weakly
coupled
with
each
other
. The
interaction
among
them
is only
established
through
the
uniqueness
of
the
macroscopic
dri
ving
force
which
are
shared,
via
the
localization
process,
by
all
the
unit
processes.
In
this
section,
we
introduce
the
set
of
controlling
unit
processes
which
have been
identified
for
describ-
ing
the
mechanical
response
of
high-purity
BCC
single
crystals,
in
particular
for
Tantalum.
We also
pro
vide
the
final
expression
resulting
from
the
the
modeling
of
each
of
these
processes.
A detailed
description
of
the
model,
including
comparison
with
experimental
data
is given
in
[9].
2.1
Dislocation
Mobility:
Double-Kink
Formation
and
Thermally
Acti
vated
Motion
of
Kinks
We consider
the
thermally
acti
vated
motion
of
dislocations
within
an
obstacle-fr
ee
slip
plane.
Under
these
conditions,
the
motion
of
the
dislocations
is
dri
ven
by
an
applied
resolv
ed
shear
stress
and
is
hindered
by
the
lattice
resistance,
which
is weak
enough
that
it may
be
overcome
by
thermal
acti
vation.
The
lattice
resistance
is presumed
to
be
well-described
by
a Peierls
ener
gy
function,
which
assigns
an
ener
gy
per
unit
length
to
dislocation
segments
as
a function
of
their
position
on
the
slip
plane.
3
A.
M.
Cuiti
̃
no
et al.
A Multiscale
Approach
for
Modeling
Crystalline
Solids
0
0.05
0.1
0.15
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
!
#"
$
!
%"
&
!
"
'
!
"
(
)
*,+
!
#"
-
.0/
1
243
Figure
2:
Temperature
dependence
of
the
effecti
ve Peierls
stress
for
various
strain
rates.
Note
that
the
typical
order
of
magnitude
of
5
687:9<;=7
>
? @0ACBEDCFHG
.
In
bcc
crystals,
the
core
of
scre
w dislocation
segments
relax
es
into
low-ener
gy
non-planar
configurations
[10
, 11
, 12
, 13
, 14
, 5, 15
, 16
]. This
introduces
deep
valle
ys
into
the
Peierls
ener
gy
function
aligned
with
the
Bur
gers
vector
directions
and
possessing
the
periodicity
of
the
lattice.
At
low temperatures,
the
dislocations
tend
to
adopt
low-ener
gy
configurations
and,
consequently
, the
dislocation
population
predominantly
con-
sists
of
long
scre
w
segments.
In
order
to
mo
ve a scre
w
segment
normal
to
itself,
the
dislocation
core
must
first
be
constricted,
which
requires
a substantial
supply
of
ener
gy. Thus,
the
ener
gy
barrier
for
the
motion
of
scre
w segments,
and
the
attendant
Peierls
stress,
may
be
expected
to
be
lar
ge,
and
the
ener
gy
barrier
for
the
motion
of
edge
segments
to
be
comparati
vely
smaller
. For
instance,
Duesbery
and
Xu
[17
] have calculated
the
Peierls
stress
for
a rigid
scre
w
dislocation
in
Mo
to
be
0.022
I
, where
I
is
the
J
@C@C@%K
shear
modulus,
whereas
the
corresponding
Peierls
stress
for
a rigid
edge
dislocation
is 0.006
I
, or
about
one
fourth
of
the
scre
w value.
This
suggests
that
the
rate-limiting
mechanism
for
dislocation
motion
is the
thermally
acti
vated
motion
of
kinks
along
scre
w segments
([18
, 19
, 20
]).
At
suf
ficiently
high
temperatures
and
under
the
application
of
a resolv
ed
shear
stress
LNM
A
, a double-
kink
may
be
nucleated
with
the
assistance
of
thermal
acti
vation
(e.
g.,
[21
, 22
, 5],
and
the
subsequent
motion
of
the
kinks
causes
the
scre
w segment
to
effecti
vely
mo
ve forw
ard,
Fig.
1.
Under
this
conditions
the
follo
wing
expression
for
the
effecti
ve temperature
and
strain-rate
dependent
Peierls
LPO
is obtained:
L0Q
?
L
>
RTS
7:9<;=78UCVXWZY\[
]
5
6
5
6
7:9<;=7
>
^0_a`cb d
efb0g
(1)
4
A.
M.
Cuiti
̃
no
et al.
A Multiscale
Approach
for
Modeling
Crystalline
Solids
da
s
screw segment
obstacle
da
e
l
e
l
s
edge segment
obstacle
Figure
3:
Bo
w-out
mechanism
for
a dislocation
segment
bypassing
an
obstacle
pair
where
the
effecti
ve Peierls
stress
is given
by
h0ikj
lnmpo<q=m
rps
m:o<q=m%tvu
(2)
and
the
reference
strain
is defined
as
w
x
m:o<q=m
i
jzy
rp{
tZuE|C}
(3)
In
the
preceding
equations,
r
is the
Bur
gers
vector
,
{
is the
dislocation
density
,
~
j%a
,
is Boltz-
mann’
s constant,
is the
absolute
temperature,
and
|
}
is the
attempt
frequenc
y which
may
be
identified
with
the
Debye
frequenc
y to
a first
approximation.
Also,
tZu
is the
distance
between
two consecuti
ve Peierls
valle
ys.
For
bcc
crystals,
tvu
j
y a
if the
slip
plane
is
C0
,
tvu
j
ya
, if the
slip
plane
is
Cy
, and
tZu
j
aC
if the
slip
plane
is
yC
, where
is the
cubic
lattice
size
[23
]. Finally
,
lm:o<q=m
is the
ener
gy
of
formation
of
a kink-pair
and
s
m:o<q=m
is the
length
of
an
incipient
double
kink.
The
formation
ener
gy
l
mpo<qm
and
the
length
s
mpo<qm
, which
cannot
be
reliably
estimated
from
elasticity
since
the
ener
gy
is composed
mostly
of
core
region,
can,
howe
ver, be
accurately
computed
by
recourse
to
atomistic
models
as
sho
wn
in
section
3.
Modeling
of
this
first
unit
process
renders
the
first
2 material
properties
amenable
of
atomistic
calculations.
In
Figure
2 the
dependence
of
the
effecti
ve Peierls
stress
on
temperature
and
rate
of
deformation
is
illustrated.
The
Peierls
stress
decreases
ostensibly
linearly
up
to
a critical
temperature
, beyond
which
it
tends
to
zero.
These
trends
are
in
agreement
with
the
experimental
observ
ations
of
Wasserb
̈
ach
[24
] and
Lachenmann
and
Schultz
[25
].
The
critical
temperature
increases
with
the
strain
rate.
In
particular
,
in
this
model
the
effect
of
increasing
(decreasing)
the
strain
rate
has
an
analogous
effect
to
decreasing
(increasing)
the
temperature,
and
vice-v
ersa,
as
noted
by
Tang
et al.
[26
]. In
the
regime
of
very
high
strain-
rates
(
w
x
0CECH
), effects
such
as
electron
and
phonon
drag
become
important
and
control
the
velocity
of
dislocations
[27
, 28
].
5
A.
M.
Cuiti
̃
no
et al.
A Multiscale
Approach
for
Modeling
Crystalline
Solids
: ¡£¢4¤#¥¦¡p§a ̈
©«ªP¬4#® ̄
¤#
°f¡¢4¤#¥¦¡p§a ̈
©ªX¬#4® ̄
¤#
±
4©¥¦²:³ ̄
4P¥¦©¡p ́μ©c¶4¬4©
® ¤8· ̧¤#²¹ º¤#¥ ̧»k¡£® ̄
¤#
¼«½=¾ ¿À½ÂÁ,Ã4ÄX½CÀ£ÅX½CÆPÄ:Á,¿Ã
Ç
¾vÄ:½CÀ«ÁÃ4ÄX½ ÀÅX½ ÆPÄ:Á¿Ã
Figure
4:
Schematic
of
ener
gy
variation
as
a function
of
a reaction
coordinate
during
dislocation
intersection
and
crossing.
2.2
Dislocation
Interactions:
Obstacle-P
air
Str
ength
and
Obstacle
Str
ength
In
the
forest-dislocation
theory
of
hardening,
the
motion
of
dislocations,
which
are
the
agents
of
plastic
deformation
in
crystals,
is impeded
by
secondary
–or
‘forest’–
dislocations
crossing
the
slip
plane.
As
the
mo
ving
and
forest
dislocations
intersect,
the
y form
jogs
or
junctions
of
varying
strengths
[29
, 30
, 4,
31
,
32
, 33
, 34
, 35,
36
] which,
pro
vided
the
junction
is suf
ficiently
short,
may
be
idealized
as
point
obstacles.
Mo
ving
dislocations
are
pinned
down
by
the
forest
dislocations
and
require
a certain
ele
vation
of
the
applied
resolv
ed
shear
stress
in
order
to
bow
out
and
bypass
the
pinning
obstacles.
For
the
case
of
infinitely
strong
obstacles,
the
resistance
of
the
forest
is pro
vided
by
the
strength
of
the
obstacle
pairs.
This
obstacle
pair
strength
is subsequently
reduced
by
considering
that
point
obstacles
composing
the
pair
can
only
pro
vide
a
finite
strength.
The
processes
imparting
the
pair
-obstacle
strength
and
obstacle
strength
are
described
next
2.2.1
Obstacle-P
air
Str
ength
We begin
by
treating
the
case
of
infinitely
strong
obstacles.
In
this
case,
pairs
of
obstacles
pin
down
dislo-
cation
segments,
which
require
a certain
threshold
resolv
ed
shear
stress
È
in
order
to
overcome
the
obstacle
pair
. The
lowest-ener
gy
configuration
of
unstressed
dislocation
segments
spanning
an
obstacle
pair
is a step
of
the
form
sho
wn
as
the
thin
line
in
Fig.
3.
Under
these
conditions,
the
bow-out
mechanism
by
which
a dislocation
segment
bypasses
an
obstacle
pair
may
be
expected
to
result
in
the
configuration
sho
wn
in
Fig.
3 (bold
line).
If the
edge-se
gment
length
is
ÉvÊ
, a displacement
ËÌÊ
of
the
dislocation
requires
a supply
of
ener
gy
equal
to
ÍÎÐϺÑμÒvÓμÔÕËÌ
Ê ÖØ×Ù
ÓμÚ=ÛPÓ
Ü
É
Ê
ËÌ
Ê
in
order
to
overcome
the
Peierls
resistance
Ù
Ó ÚÛPÓ
Ü
and
to
ex-
tend
the
scre
w
segments.
The
corresponding
ener
gy
release
is
×Ù
É
Ê
ËÌ
Ê
. Similar
contrib
utions
result
from
a
displacement
ËÌÝ
of
the
scre
w-se
gment
of
length
ɺÝ
. Retaining
dominant
terms
the
obstacle-pair
strength
is
ÈßÞ
Ù
ϺÑμÒvÓμÔ
Ü
Ö
ÍÎÐÓ ÚÛPÓ
×
ɺÝ
(4)
The
obstacle-pair
strength
can
be
therefore
estimated
by
quantifying
Ù
Ü
,
ÉvÝ
and
ÎàÓμÚ=ÛPÓ
. An
expression
for
the
Peierls
stress
Ù
Ü
is given
in
Eq.
(1),
the
distance
between
obstacles
along
the
scre
w direction
ɺÝ
is estimated
by
statistics
assuming
a random
obstacle
distrib
ution
and
the
core
ener
gy
per
unit
length
in the
edge
direction
ÎÐÓ ÚÛPÓ
is obtained
by
atomistic
calculations
presented
in
the
follo
wing
sections.
6