MAGNETIZED FIBER ORIENTATION AND CONCENTRATION CONTROL IN SOLIDIFYING COMPOSITES

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16 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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MAGNETIZED FIBER ORIENTATION AND
CONCENTRATION CONTROL IN
SOLIDIFYING COMPOSITES





GEORGE S. DULIKRAVICH and
Marcelo J. Colaço

University of Texas at Arlington, Mechanical and Aerospace Eng. Dept., MAIDO Institute;

UTA Box 19018; Arlington, TX 76019, USA


dulikra@mae.uta.edu


THOMAS J. MARTIN

Pratt & Whitney Engine Company, Turbine Discipline Engineering & Optimization Group;

400 Main Street, M/S 169
-
20; East Hartford, CT 06108, USA



Seungsoo Lee

Department of Aerospace Engineering, Inha University, Inchon, Korea



Defects in short fiber composites
are often due to uncontrolled
fiber orientation and
concentration during composites
manufacturing

In many applications it would be
highly desirable to have
directional dependence of
physical properties of the
material, that is, to have strongly
non
-
isotropic materials.

It would be of interest to perform
curing of the resin in such a way
that the local concentration and
orientation of the fibers is fully
controlled.

During the solidification process, melt flow is
generated due to strong thermal buoyancy
forces. This process cannot be effectively
controlled in the case of strong heat transfer,
except if influenced by a global body force.
One such body force is the general
electromagnetic Lorentz force that is created
in any electrically conducting fluid when
either a magnetic field or an electric field is
applied.


During the curing process in composites
manufacturing, we usually work with
electrically conducting liquid polymers
and carbon fibers, although a variety of
other molten substances and fibers made
of other materials are often used.

The resins are electrically conducting
either because of the presence of iron
atoms, salts, or acids.


If short carbon fibers (5
-
10 microns in
diameter and 200 microns long) are vapor
-
coated with a thin layer (2
-
3 microns) of a
ferromagnetic material like nickel, the fibers
will respond to the externally applied
electromagnetic fields by rotating and
translating so that they become aligned with
the magnetic lines of force (Hatta and
Yamashita, 1988; Yamashita et al., 1989).


The objective of this work is to explore
the feasibility of manufacturing specialty
metal matrix and polymer composite
materials that will have specified
(desired) locally directional variation of
bulk physical properties like thermal and
electrical conductivity, modulus of
elasticity, thermal expansion coefficient,
etc.


The fundamental concept is based on
specifying a desired pattern of orientations
and spacing of micro fibers in the final
composite material product.

Then, the task is to determine the proper
strengths, locations, and orientations of
magnets that will have to be placed along the
boundaries of the curing composite part so
that the resulting magnetic field lines of force
will coincide with the specified (desired)
pattern of the micro fibers’ distribution.


The basic idea is that the fibers will align
with the local magnetic lines of force
(Hatta and Yamashita, 1988).


It is important to understand that the
pattern of these lines depends on the
solidifying resin thermally and
magnetically influenced flow
-
field and
the spatial variation of the applied
magnetic field.

The successful proof of this
manufacturing concept involves the
development of an appropriate software
package for the numerical solution of the
partial differential equation system
governing magneto
-
hydro
-
dynamics
(MHD) involving combined fluid flow,
magnetic field, and heat transfer that
includes liquid
-
solid phase change
(Dulikravich, 1999).


It also involves development of a
constrained optimization software that is
capable of automatically determining the
correct strengths, locations, and
orientations of a finite number of
magnets that will produce the magnetic
field force pattern which coincides with
the desired and specified fiber
concentration and orientation pattern in
the curing composite material part.

A Mathematical Model of Magneto
-
Hydro
-
Dynamics (MHD) With Solidification


n
θ
n
T
T
T
T
V
V
V
f
solid
liquid
solid
s
~
























r
r
r
r
r
r
1
1

































s
mix
f)
(1

f















d
θ
df
L
c
c
T
T
c
f
-
1
fc
c
s
s
s
mix








Non
-
dimensional numbers

vr
r
r
r
v




Re
r
r
vr
c
Pr



r
r
r
T
c
v
Ec


2
2
3
2
vr
r
r
r
r
r
T
g
Gr






r
r
r
vr
Pm





2
/
1
vr
r
r
r
r
H
Ht













Mass and momentum conservation

0



v













































































H
H
g
v
v
H
H
g
v
v
I
vv
I
vv
s
s
s
T
T
s












2
2
2
vs
2
2
2
v
s
Re
Pm
Ht
Re
Gr
Re
f)
(1
PmRe
Ht
Re
Gr
Re
f
p
f
1
p
f








H
H











2
2
2
Re
Pm
Ht
Fr
p
p
H
H




s
2
2
2
s
s
Re
Pm
Ht
Fr
p
p







g
Energy conservation and

magnetic field transport






























































H
H
H
H
v
mix
3
e
2
m
c
2
t
s
s
r
e
3
e
2
m
c
2
t
r
e
mix
R
P
E
H
1
P
R
1
f)
(1
R
P
E
H
1
P
R
1
f
c


0



B
J
H
=


0




J
B
v
J










H
H
v
2
s
s
PmRe
f)
(1
f














Geometry and boundary conditions
for test cases 1 and 2.


g
0
=9.81 m/s
2

T
c

T
h

Insulated

Insulated

B
4

(
x
)

B
3

(
x
)

B
1

(
y
)

B
2

(
y
)

Discretization of the magnetic field
strength along the boundaries





2
/
1
cells
#
1
i
2
calculated
y
specified
y
cells
#
1
i
2
calculated
x
specified
x
B
B
cells
#
1
B
B
cells
#
1
F























M
1
i
k
i
i
k
x
C
P
x
B


1,3,5,...
i
for

x
2
)
1
i
(
cos
x
C
k
k
i












2,4,6,...
i
for

x
2
i
cos
x
C
k
k
i









Test case 1.

In the inverse problem of determining
the unknown magnetic field boundary
conditions in test case 1 we used
three

parameters for
B
1
(
y
) and
three

parameters for
B
3
(
x
), while magnetic
boundary conditions on the opposite
walls were enforced as
periodic
, that is
B
2
(1,
y
) =
B
1
(0,
y
) and
B
4
(
x,1
) =
B
3
(
x,0
).


Specified and estimated boundary conditions for
magnetic field in test case 1.


0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
y
-
0
.
0
8
-
0
.
0
4
0
.
0
0
0
.
0
4
0
.
0
8
B
x

/

B
o
E
s
t
i
m
a
t
e
d
P
r
e
s
c
r
i
b
e
d
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
x
0
.
1
2
0
.
1
4
0
.
1
6
0
.
1
8
B
y

/

B
o
E
s
t
i
m
a
t
e
d
P
r
e
s
c
r
i
b
e
d
Specified and calculated magnetic and
field distributions for test case 1.


Specified and calculated temperature
field distributions for test case 1.


-
0
.
4
4
-
0
.
4
4
-
0
.
3
8
-
0
.
3
8
-
0
.
3
1
-
0
.
3
1
-
0
.
2
5
-
0
.
2
5
-
0
.
2
5
-
0
.
1
9
-
0
.
1
9
-
0
.
1
9
-
0
.
1
3
-
0
.
1
3
-
0
.
0
6
-
0
.
0
6
-
0
.
0
6
0
.
0
0
0
.
0
0
0
.
0
6
0
.
0
6
0
.
0
6
0
.
1
3
0
.
1
3
0
.
1
9
0
.
1
9
0
.
1
9
0
.
2
5
0
.
2
5
0
.
3
1
0
.
3
1
0
.
3
1
0
.
3
8
0
.
3
8
0
.
4
4
0
.
4
4
0
.
4
4
-
0
.
4
4
-
0
.
4
4
-
0
.
3
8
-
0
.
3
8
-
0
.
3
8
-
0
.
3
1
-
0
.
3
1
-
0
.
3
1
-
0
.
2
5
-
0
.
2
5
-
0
.
2
5
-
0
.
1
9
-
0
.
1
9
-
0
.
1
3
-
0
.
1
3
-
0
.
1
3
-
0
.
0
6
-
0
.
0
6
-
0
.
0
6
0
.
0
0
0
.
0
0
0
.
0
0
0
.
0
6
0
.
0
6
0
.
1
3
0
.
1
3
0
.
1
9
0
.
1
9
0
.
1
9
0
.
2
5
0
.
2
5
0
.
2
5
0
.
3
1
0
.
3
1
0
.
3
8
0
.
3
8
0
.
4
4
0
.
4
4
0
.
4
4
Convergence history for the hybrid
optimization for test case 1.


0
2
0
4
0
6
0
I
t
e
r
a
t
i
o
n

n
u
m
b
e
r
0
.
0
0
0
.
0
2
0
.
0
4
0
.
0
6
0
.
0
8
C
o
s
t

f
u
n
c
t
i
o
n
S
Q
P
D
F
P
G
A
N
M
0
2
0
0
4
0
0
6
0
0
8
0
0
#

o
f

o
b
j
.

f
u
n
c
.

e
v
a
l
.
L
M
D
E
Test case 2.

In this case,
B
1
(0,
y
) was approximated with
only
six

parameters although the actual value
was a discontinuous function. The value of
B
3
(
x,0
) was similarly approximated with
six

parameters although the actual value was a
constant. Magnetic boundary conditions on
the opposite walls were enforced as
periodic
,
that is
B
2
(1,
y
) =
B
1
(0,
y
) and
B
4
(
x,1
) =
B
3
(
x,0
).


Specified and estimated boundary
conditions for magnetic field in test case 2.


0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
y
0
.
4
0
.
8
1
.
2
1
.
6
2
.
0
B
x

/

B
o
E
s
t
i
m
a
t
e
d
P
r
e
s
c
r
i
b
e
d
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
x
0
.
0
E
+
0
4
.
0
E
-
3
8
.
0
E
-
3
1
.
2
E
-
2
1
.
6
E
-
2
B
y

/

B
o
E
s
t
i
m
a
t
e
d
P
r
e
s
c
r
i
b
e
d
Specified and calculated magnetic and
field distributions for test case 2.

Specified and calculated temperature
field distributions for test case 2.

-
0
.
4
4
-
0
.
4
4
-
0
.
3
8
-
0
.
3
8
-
0
.
3
8
-
0
.
3
1
-
0
.
3
1
-
0
.
3
1
-
0
.
2
5
-
0
.
2
5
-
0
.
2
5
-
0
.
1
9
-
0
.
1
9
-
0
.
1
3
-
0
.
1
3
-
0
.
1
3
-
0
.
0
6
-
0
.
0
6
-
0
.
0
6
0
.
0
0
0
.
0
0
0
.
0
0
0
.
0
6
0
.
0
6
0
.
1
3
0
.
1
3
0
.
1
9
0
.
1
9
0
.
1
9
0
.
2
5
0
.
2
5
0
.
2
5
0
.
3
1
0
.
3
1
0
.
3
8
0
.
3
8
0
.
3
8
0
.
4
4
0
.
4
4
0
.
4
4
-
0
.
4
4
-
0
.
4
4
-
0
.
3
8
-
0
.
3
8
-
0
.
3
1
-
0
.
3
1
-
0
.
2
5
-
0
.
2
5
-
0
.
2
5
-
0
.
1
9
-
0
.
1
9
-
0
.
1
9
-
0
.
1
3
-
0
.
1
3
-
0
.
0
6
-
0
.
0
6
-
0
.
0
6
0
.
0
0
0
.
0
0
0
.
0
6
0
.
0
6
0
.
0
6
0
.
1
3
0
.
1
3
0
.
1
3
0
.
1
9
0
.
1
9
0
.
1
9
0
.
2
5
0
.
2
5
0
.
2
5
0
.
3
1
0
.
3
1
0
.
3
1
0
.
3
8
0
.
3
8
0
.
4
4
0
.
4
4
0
.
4
4
Convergence history for the hybrid
optimization for test case 2.

0
2
0
4
0
6
0
8
0
I
t
e
r
a
t
i
o
n

n
u
m
b
e
r
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
C
o
s
t

f
u
n
c
t
i
o
n
S
Q
P
D
F
P
G
A
N
M
0
2
0
0
4
0
0
6
0
0
8
0
0
1
0
0
0
#

o
f

o
b
j
.

f
u
n
c
.

e
v
a
l
.
L
M
D
E
Geometry and boundary conditions
for test case 3.

g
0
=9.81 m/s
2

T
c

T
h
(
x
)

Insulated

Insulated

x=0

x=1

B
1

(
y
)

B
2

(
y
)

B
4

(
x
)

B
3

(
x
)

Test case 3.

In this case,
three

separate parameters were
used to parameterize each of the
four

magnetic boundary conditions, where and .
That is, magnetic field boundary conditions
were
not

explicitly treated as periodic.

Vertical walls were adiabatic, top wall was
isothermal, bottom wall had quadratically
varying symmetric temperature distribution.


Specified and estimated boundary
conditions for magnetic field in test case 3.


0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
y
-
2
.
0
E
-
5
-
1
.
0
E
-
5
0
.
0
E
+
0
1
.
0
E
-
5
2
.
0
E
-
5
B
x

/

B
o
E
s
t
i
m
a
t
e
d
P
r
e
s
c
r
i
b
e
d
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
x
0
.
8
1
.
2
1
.
6
2
.
0
B
y

/

B
o
E
s
t
i
m
a
t
e
d
P
r
e
s
c
r
i
b
e
d
Specified and calculated magnetic and
field distributions for test case 3.

Specified and calculated temperature
field distributions for test case 3.

-
0
.
4
3
-
0
.
4
3
-
0
.
3
6
-
0
.
3
6
-
0
.
2
9
-
0
.
2
9
-
0
.
2
2
-
0
.
2
2
-
0
.
1
6
-
0
.
1
6
-
0
.
0
9
-
0
.
0
9
-
0
.
0
2
-
0
.
0
2
0
.
0
5
0
.
0
5
0
.
1
2
0
.
1
2
0
.
1
9
0
.
1
9
0
.
1
9
0
.
2
6
0
.
2
6
0
.
3
3
0
.
3
3
0
.
3
9
0
.
3
9
0
.
4
6
0
.
4
6
0
.
5
3
0
.
5
3
-
0
.
4
3
-
0
.
4
3
-
0
.
3
6
-
0
.
3
6
-
0
.
2
9
-
0
.
2
9
-
0
.
2
2
-
0
.
2
2
-
0
.
1
6
-
0
.
1
6
-
0
.
0
9
-
0
.
0
9
-
0
.
0
2
-
0
.
0
2
0
.
0
5
0
.
0
5
0
.
1
2
0
.
1
2
0
.
1
9
0
.
1
9
0
.
2
6
0
.
2
6
0
.
2
6
0
.
3
3
0
.
3
3
0
.
3
9
0
.
3
9
0
.
4
6
0
.
4
6
0
.
5
3
0
.
5
3
Convergence history for the hybrid
optimization for test case 3
.

0
1
0
2
0
3
0
4
0
I
t
e
r
a
t
i
o
n

n
u
m
b
e
r
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
C
o
s
t

f
u
n
c
t
i
o
n
S
Q
P
D
F
P
G
A
N
M
0
4
0
0
8
0
0
1
2
0
0
1
6
0
0
#

o
f

o
b
j
.

f
u
n
c
.

e
v
a
l
.
L
M
D
E
DISCUSSION

In this study, a number of assumptions were made concerning physics of
the problem. For example, all physical properties were treated as constants
instead of as functions of temperature.

Transport equation for the passive scalar (concentration of the micro
fibers) was not included, but could be added relatively easily (Hirtz and
Ma, 2000). Its addition would enable us to predict and possibly control the
distribution of the micro
-
fibers along the magnetic field lines of force.
Thermal stresses in the accrued solid were not analyzed since such solids
are by definition non
-
isotropic.

Different options for treating the mushy region were not exercised (Poirier
and Salcudean, 1986).

Other, possibly more robust and accurate numerical integration methods
were not explored (Fedoseyev et al., 2001; Dennis and Dulikravich, 2002)
that could allow for physical values of the magnetic Prandtl number and
for significantly higher values of viscosity used in the solid region.


Magnetization effects and fiber
-
resin interface drag were neglected.
Also, possible effects of the material properties and the thickness
and shape of the container walls were not included via a conjugate
analysis (Dennis and Dulikravich, 2000).


Finally, the current effort neglects the fact that the entire problem of
having magnetizable micro
-
fibers orient themselves tangent to the
prescribed magnetic field pattern is feasible only if the prescribed
pattern is enforced in the moving and deforming mushy region.


This mandates that the entire problem should in reality be treated as
an unsteady control problem where boundary values of the magnetic
field should vary in time.


All of these details could and should be incorporated in the future
work and compared to actual experimental results since this concept
could be extended to manufacturing of three
-
dimensional composite
objects and functionally graded objects of arbitrary shape.

Summary


Feasibility

of

a

new

concept

has

been

demonstrated

for

manufacturing

composite

materials

where

micro

fibers

will

align

along

a

user
-
specified

desired

pattern

of

the

magnetic

lines

of

force
.



This

was

accomplished

by

combining

an

MHD

analysis

code

capable

of

simultaneously

capturing

features

of

the

melt

flow
-
field

and

the

accrued

solid,

and

a

hybrid

constrained

optimization

code
.



The

computed

pattern

of

the

magnetic

lines

of

force

was

shown

to

closely

replicate

the

specified

pattern

when

the

optimizer

minimized

the

L
2
-

norm

of

the

difference

between

these

two

patterns
.



This

minimization

process

was

achieved

by

optimizing

a

finite

number

of

parameters

describing

analytically

the

distribution

and

the

orientations

of

the

boundary

values

of

the

magnetic

field
.