Calculation of Linear and Non-Linear Elastic Properties of Polycrystalline Materials by Use of a Cluster Model

Abstract

In the present paper we have put together some results of a cluster method that allows the calculation
of linear and also non-linear effective elastic constants of polycrystalline materials within an iterative
self-consistent scheme. The conceptual idea consists in simulating the real material by a suitably chosen
cluster of single grains. One can then determine the elastic properties of the material under study by
examining the elastic behavior of the cluster. The method is capable of resolving the effect of the grain
shape, that is determined by the coordination number of the grains on the effective constants.

Full text

Textures
and
Microstructures,
1997,
Vol.
28,
pp.
261-271
Reprints
available
directly
from
the
publisher.
Photocopying
permitted
by
license
only
(C)
1997
OPA
(Overseas
Publishers
Association)
Amsterdam
B.V.
Published
in
The
Netherlands
under
license
by
Gordon
and
Breach
Science
Publishers
Printed
in
Malaysia
CALCULATION
OF
LINEAR
AND
NON-LINEAR
ELASTIC
PROPERTIES
OF
POLYCRYSTALLINE
MATERIALS
BY
USE
OF
A
CLUSTER
MODEL
H.
KIEWEL
1,
H.
J.
BUNGE
2
and
L.
FRITSCHE
llnstitute
of
Theoretical
Physics
B
2Department
of
Physical
Metallurgy,
Technical
University
Clausthal,
Germany
(Received
18
April
1996)
In
the
present
paper
we
have
put
together
some
results
of
a
cluster
method
that
allows
the
calculation
of
linear
and
also
non-linear
effective
elastic
constants
of
polycrystalline
materials
within
an
iterative
self-consistent
scheme.
The
conceptual
idea
consists
in
simulating
the
real
material
by
a
suitably
chosen
cluster
of
single
grains.
One
can
then
determine
the
elastic
properties
of
the
material
under
study
by
examining
the
elastic
behavior
of
the
cluster.
The
method
is
capable
of
resolving
the
effect
of
the
grain
shape,
that
is
determined
by
the
coordination
number
of
the
grains
on
the
effective
constants.
KEY
WORDS:
Effective
elastic
constants,
cluster
method,
grain
shape.
INTRODUCTION
The
prediction
of
effective
elastic
constants
of
polycrystalline
materials
from
properties
of
the
constituting
individual
grains
and
their
interaction
presents
a
frequently
appearing
problem.
First
attempts
to
solve
this
problem
were
made
by
Voigt
(1910)
and
Reuss
(1929).
These
authors
assume
that
throughout
the
entire
material
the
strain
and
stress,
respectively,
are
constant
which
is
in
far
contrast
with
well-known
experimental
facts.
Their
most
important
meaning
consists
in
yielding
upper
and
lower
bounds,
respectively,
of
the
actual
polycrystal
data
(Hill,
1952).
The
self-consistent
approach
by
Kr6ner
(Kr6ner,
1958;
Kneer,
1964;
Kneer,
1965;
Morris,
1971)
was
the
first
successful
method
of
taking
the
grain
interaction
into
account.
If
the
moduli
in
space
are
perfectly
disordered
Kr6ner’s
value
is
identical
with
the
actual
polycrystal
data
(Kr6ner,
1977).
The
well-known
bounds
of
Hashin
and
Shtrikman
(1962a,
b)
are
much
closer
than
Hill’s
bounds,
but
they
are
only
valid
if
there
is
no
correlation
between
the
elastic
moduli
of
neighboring
volume
elements
inside
the
material.
In
the
seventies
all
these
different
values
for
the
effective
moduli
were
classified
after
their
order
of
correlation
(Zeller
and
Dederichs
(1973);
Kr6ner
(1977)).
In
the
past
decade
the
vast
increase
in
the
capacity
of
modem
computers
has
led
to
a
new
access
to
determine
material
properties.
New
methods
were
developed
which
simulate
the
arrangement
of
grains
inside
a
polycrystal
by
a
cluster
of
500
to
1000
grains
(Kumar
1992;
Kiewel
and
Fritsche
(1994a,
1994b)).
261

262
H.
KIEWEL
ET-AL.
METHOD
Here,
we
only
give
a
brief
description
of
the
used
cluster
approach
(for
details
see
Kiewel
and
Fritsche
(1994b)
and
Kiewel,
Fdtsche
and
Reinert
(1995)).
The
conceptual
idea
consists
in
simulating
the
real
material
by
a
suitably
chosen
cluster
of
single
grains.
The
cluster
is
embedded
in
a
homogeneous
medium
that
has
approximately
the
effective
elastic
constants
of
the
material
under
study.
To
determine
the
elastic
properties
we
subject
to
the
entire
surface
of
the
cluster
a
homogeneous
deformation.
In
order
to
determine
the
deformation
field
inside
the
aggregate,
one
has
to
solve
the
associated
boundary
value
problem.
To
this
end
we
expand
the
displacement
field
inside
the
individual
grains
in
terms
of
basis
functions
which
fulfill
separately
the
time-independent
fundamental
equation
of
elasticity.
The
expansion
coefficients
are
obtained
from
minimizing
the
mean
square
mismatch
of
the
displacement
and
the
stress
at
the
grain
boundaries.
If
one
then
averages
the
strain
and
stress
tensor
over
the
entire
cluster
and
employs
a
general
stress-strain
relation
for
the
polycrystalline
material
one
finally
obtains
linear
and
also
non-linear
effective
elastic
constants.
RESULTS
AND
DISCUSSION
To
gain
a
first
impression
of
the
elastic
properties
of
polycrystalline
materials
we
have
calculated
the
deformation
field
inside
three
different
clusters.
For
clarity,
each
cluster
consists
of
only
one
type
of
grain
shape.
We
have
chosen
Wigner-Seitz
cells
for
the
grain
shape
to
fill
out
the
entire
space
and
to
avoid
overlapping.
Our
construction
scheme
of
the
aggregate
starts
with
one
grain
in
the
origin
which
is
surrounded
by
nearest
neighbors,
next
nearest
neighbors,
and
so
on.
For
that
reason
the
shape
of
the
cluster
is
approximately
spherical,
since
the
grains
are
arranged
in
shells.
The
first
type
of
cluster
consists
of
365
Wigner-Seitz
cells
of
a
simple
cubic
(sc)
lattice,
the
second
one
of
181
cells
of
a
body
centered
cubic
(bcc)
lattice
and
the
third
one
of
201
cells
of
a
face
centered
cubic
(fcc)
lattice.
These
particular
numbers
of
cells
are
due
to
the
arrangement
in
shells.
For
different
grains
the
orientation
of
the
lattice
should
be
completely
uncorrelated.
Each
cluster
is
subjected
to
a
uniaxial
strain
at
its
surface.
The
resulting
relative
changes
of
the
specific
volume
V(r)
Tr
(())
()
and
the
displacement
_u(_r)
uo(_r),
(2)
where
u0__(r)
denotes
the
given
homogeneous
displacement
at
the
surface
of
the
cluster
are
plotted
in
Figures
1
to
3.
For
a
medium
with
discontinuous
elastic
properties
the
strain
has
to
be
discontinuous
since
the
stress
is
always
continuous.
As
a
consequence
the
field
of
the
relative
changes
of
the
volume
is
discontinuous,
which
is
apparent
from
Figures
l(a)
to
3(a).
At
the
grain
boundaries
the
displacement
has
to
be
continuous
which
is
confirmed
by
Figures
l(b)
to
3(b)
since
it
is
impossible
to
resolve
any
grain
boundary.
In
the
range
of
the
applied
displaying
technique
the
displacerent
field
is
continuous.

ELASTIC
PROPERTIES
263
Figure
1
(a)
Relative
changes
of
the
specific
volume
6V(r)
Tr
(=(r))
(1)
V0
in
equidistant
steps
inside
a
cluster
constituting
of
Wigner-Seitz
cells
of
a
simple
cubic
lattice
that
are
common
cubes.
The
cut
is
taken
through
the
center
of
the
cluster
(r
0)
along
the
plane
z
0
containing
the
strain
axis.
Because
of
the
large
number
of
grains,
we
only
display
a
narrow
region
centered
at
r
0.
(b)
The
displacement
field
u(r)-
u(r)
associated
with
the
elastically
deformed
cluster
shown
in
(a).

264
H.
KIEWEL
ET
AL.
Figure
2
(a)
Relative
changes
of
the
specific
volume
and
(b)
displacement
field
inside
a
cluster
constituting
of
Wigner-Seitz
cells
of
a
body
centered
cubic
lattice.
The
situation
is
the
same
as
in
Figure
1.

ELASTIC
PROPERTIES
265
Figure
3
(a)
Relative
changes
of
the
specific
volume
and
(b)
displacement
field
inside
a
cluster
constituting
of
Wigner-Seitz
cells
of
a
face
centered
cubic
lattice.
The
situation
is
the
same
as
in
Figure
1.

266
H.
KIEWEL
ET
AL.
After
these
introductory
investigations
we
now
calculate
effective
elastic
moduli
depending
on
the
grain
shape.
For
simplification
we
examine
only
macroscopically
isotropic
materials.
Figure
4
and
additionally
Table
1
show
the
polycrystal
constants
for
copper.
As
has
to
be,
all
moduli
lie
within
the
bounds
of
Hill
and
nearly
all
moduli
fall
within
the
bounds
of
Hashin
and
Shtdkman.
The
clusters
consist
of
a
number
of
grains
that
is
quite
small
in
comparison
with
the
great
numbers
used
for
statistical
investigations.
For
that
reason
the
size
of
the
grains
is
relatively
large
compared
with
the
size
of
the
entire
cluster.
Therefore
it
is
easy
to
understand
that
some
values
lie
a
little
outside
the
bounds
of
Hashin
and
Shtrikman,
since
the
requirement
that
there
Table
1
Results
for
the
moduli
of
macroscopically
isotropic
copper
in
GPa.
The
single
crystal
data
are
taken
from
(Bradfield,
1962).
The
columns
HS1
and
HS2
contain
the
values
of
Hashin
and
Shtrikman.
The
different
grain
shapes
of
the
cluster
method
are
labeled
by
fcc,
bcc
and
sc
(see
text).
Reuss
HS1
KrSner
Cluster
method
HS2
Voigt
fcc
bcc
sc
B
137.6
137.6
137.6
137.6
137.6
137.6
137.6
137.6
_+0.0
_-+0.0
_+0.0
G
40.1
46.1
48.3
46.6
47.6
49.5
49.5
54.7
_+0.2
_-/-0.2
_-_+0.1
E
109.7
124.3
129.7
125.6
127.9
132.6
132.7
145.0
_+0.4
_+0.5
_+0.2
14.2-
142
14.1
140
138-
1,37-
=
136"
rn
Voigt
Reuss
f
cc
bcc
sc
141
140
138
137
136
135
134
133,
133

ELASTIC
PROPERTIES
267
56-
54-
52-
o
5ff
--=
48-
o
E
46-
44-
42.
38.
Voigt
HS2
HS1
Reuss
f
cc
bcc
sc
56
54
52
5O
48
46
44
42
4O
38
150-
150
145-
140-
13,5
"5
130
o
E
125-
120"
o
115,
110"
Voigt
HS2
HS1
Reuss
f
cc
bcc
sc
145
140
135
130
125
120
115
110
105
105
Figure
4
The
elastic
moduli
of
macroscopically
isotropic
copper
metal
in
GPa.
We
label
the
bounds
of
Hill
by
Reuss
and
Voigt.
The
symbols
HS1
and
HS2
stand
for
the
lower
and
upper
bounds
of
Hashin
and
Shtrikman
(1962a,b),
respectively.
For
every
type
of
grain
shape
we
have
chosen
10
different
sets
of
random
orientations
of
the
crystal
lattice
inside
the
grains.
The
results
for
every
set
are
displayed
by
a
short
bar.
We
sign
the
different
grain
shapes
by
the
common
abbreviations
for
the
Wigner-Seitz
cells
constituting
the
cluster:
sc
(simple
cubic),
bcc
(body
centered
cubic),
fcc
(face
centered
cubic).

268
H.
KIEWEL
ET
AL.
is
no
correlation
between
neighboring
volume
elements
can
only
approximately
be
fulfilled.
The
maximum
difference
of
the
effective
moduli
for
different
grain
shapes
is
about
6%
of
the
average
value
for
the
three
investigated
types
of
cluster.
In
Figure
5
and
Table
2
the
results
for
graphite
are
presented.
The
values
of
Voigt
and
Reuss
and
even
the
bounds
of
Hashin
and
Shtrikman
vary
enormously
since
single
grains
of
graphite
are
extremely
anisotropic.
Therefore
the
effective
elastic
constants
depend
crucially
on
the
grain
shape.
It
is
therefore
absolutely
necessary
for
the
calculation
of
Table
2
Results
for
the
moduli
of
macroscopically
isotropic
graphite
in
GPa.
The
single
crystal
data
are
taken
from
(Landolt-B/Srnstein,
1979).
The
columns
HS1
and
HS2
contain
the
values
of
Hashin
and
Shtrikman.
The
different
grain
shapes
of
the
cluster
method
are
labeled
by
fcc,
bcc
and
sc
(see
text).
Reuss
HS1
Kr6ner
Cluster
method
HS2
Voigt
fcc
bcc
sc
B
35.8
42.0
88.5
104.9
120.1
138.4
204.2
286.3
+2.2
+3.4
+2.4
G
9.21
14.9
53.3
58.9
73.0
91.3
148.9
219.4
+0.9
+1.1
+1.2
E
25.5
39.9
133.1
148.9
182.2
224.5
359.4
524.2
+2.3
+3.0
+3.1
30ff
300
275"
250-
225"
200
175-
-o
150-
o
5-
=
100"
5ff
Voigt
HS2
HS1
Reuss
f
cc
bcc
sc
275
250
225
200
175
150
125
100
75
50
25
0
0

ELASTIC
PROPERTIES
269
250-
225-
200-
Voigt
HS2
HS1
Reuss
f
cc
bcc
sc
250
225
200
175
150
125
100
75
50
25
550"
50(
45(T
40(T
35-----
30O
25(F
20(
150
Voigt
HS2
aeHS
u$$
55O
500
450
400
350
300
25O
200
150
100
5O
0
f
cc
bcc
sc
0
Figure
5
The
elastic
moduli
of
macroscopically
isotropic
graphite
in
GPa.
The
situation
is
the
same
as
in
Figure
4.

270
H.
KIEWEL
ET
AL.
reliable
effective
constants
of
graphite
to
take
the
grain
shape
into
consideration.
The
results
for
these
two
materials
indicate
an
important
effect
which,
is
in
more
detail,
described
in
(Kiewel,
Bunge
and
Fritsche,
1995):
With
increasing
coordination
number
of
the
grains
the
elastic
moduli
decrease.
Finally
we
show
some
results
for
the
non-linear
elastic
moduli
taken
from
(Kiewel,
Fritsche
and
Reinert,
1995).
In
Tables
3
and
4
the
effective
3rd-order
stiffnesses
for
macroscopically
isotropic
aluminium
and
copper,
respectively,
are
displayed.
Especially
for
copper
there
is
a
wide
spread
in
the
experimental
values.
Therefore
they
can
hardly
serve
as
reliable
values
for
the
polycrystalline
materials
and
a
theoretical
determination
becomes
essential.
For
aluminium
which
has
a
very
small
anisotropy
of
the
individual
grains,
the
Voigt
value
which
is
the
average
over
all
orientations,
and
the
result
of
the
cluster
approach,
are
nearly
identical.
Here,
the
Voigt
value
may
serve
as
a
criterion
to
test
the
reliability
of
the
extensive
computer
code
used
for
the
cluster
method.
For
the
more
anisotropic
copper
the
Voigt
value
is
less
good
and
a
calculation
by
use
of
a
cluster
model
becomes
essential.
Table
3
Results
for
the
non-linear
effective
stiffnesses
of
macroscopically
isotropic
aluminium
in
GPa.
The
single
crystal
data
are
taken
from
(Landolt-Btimstein,
1992),
the
experimental
data
from
(Landolt-B6rnstein,
1984).
Voigt
Cluster
bcc
Experiment
Cll
-1506
-1496
-1479 -1634
C112
-302 -303 -287
-454
c23
-47
-51
-39
-204
c4
-128
-126
-124
-125
c55
-301
-298
-298
-295
c456
-87
-86
-87
-85
Table
4
Results
for
the
non-linear
effective
stiffnesses
of
macroscopically
isotropic
copper
in
GPa.
The
single
crystal
data
are
taken
from
(Landolt-B6rnstein,
1992),
the
experimental
data
from
(Landolt-Btimstein,
1984).
Voigt
Cluster
bcc
Experiment
Cll
-2470
-2187
-2745
-1753
C112
-544
-605
-313
-377
c123
-329
-288
-657
-173
c4
-107 -159
+172
-102
C155
-482
-395 -608
-344
c456
-187 -118
-390
-121

ELASTIC
PROPERTIES
271
CONCLUSIONS
The
results
of
the
present
contribution
show
that
the
cluster
method
is
a
reliable
tool
to
determine
linear
and
also
non-linear
elastic
properties
of
polycrystalline
materials.
The
accuracy
of
this
scheme
is
sufficient
to
resolve
the
influence
of
the
grain
shape
on
the
elastic
constants.
Further
investigations
will
handle
correlation
effects
between
different
grains.
Acknowledgments
The
present
work
was
financially
supported
by
the
Deutsche
Forschungsgemeinschaft
which
is
gratefully
acknowledged.
References
Bradfield,
G.
(1962).
Private
communication
to
Kneer,
G.
(quoted
in
(Kneer,
1965)).
Hashin,
Z.
and
Shtrikman,
S.
(1962a).
On
some
variational
principles
in
anisotropic
and
nonhomogeneous
elasticity,
J.
Mech.
Phys.
Solids,
1tl,
335.
Hashin,
Z.
and
Shtrikman,
S.
(1962b).
A
variational
approach
to
the
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