MAGNETIZED FIBER ORIENTATION AND CONCENTRATION CONTROL IN

SOLIDIFYING COMPOSITES

 

GEORGE S. DULIKRAVICH and Marcelo J. ColaçoUniversity of Texas at Arlington, Mechanical and Aerospace Eng. Dept., MAIDO Institute;

UTA Box 19018; Arlington, TX 76019, USA dulikra@mae.uta.edu

THOMAS J. MARTINPratt & Whitney Engine Company, Turbine Discipline Engineering & Optimization Group;

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

 Seungsoo LeeDepartment 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

nθ

n

TT

TT

VV

Vf

solidliquid

solid

s

~

rrrr

rr 11

smix f)(1 f

dθ

dfLccTTcf-1fcc sssmix

Non-dimensional numbers

vr

rrrv

Rer

rvrcPr

rr

r

Tc

vEc

2

2

32

vr

rrrrr TgGr

r

rrvrPm

2/1

vr

rrrr HHt

Mass and momentum conservation

0 v

HHgvv

HHgvv

IvvIvv

sssT

T

s

2

2

2vs

2

2

2v

s

RePm

Ht

Re

Gr

Ref)(1

PmRe

Ht

Re

Gr

Ref

pf1pf

HH

2

2

2 RePm

Ht

Fr

pp HH s2

2

2s

s RePm

Ht

Fr

pp

g

Energy conservation and magnetic field transport

HH

HHvmix

3e

2m

c2t

ss

re

3e

2m

c2t

remix

RP

EH1

PR

1f)(1

RP

EH1

PR

1fc

0 B JH = 0 J BvJ

HHv 2ss

PmRe

f)(1f

Geometry and boundary conditions

for test cases 1 and 2.

g0=9.81 m/s2Tc Th

Insulated

Insulated

B4 (x)

B3 (x)

B1 (y) B2 (y)

Discretization of the magnetic field strength along the boundaries

2/1cells#

1i

2calculatedy

specifiedy

cells#

1i

2calculatedx

specifiedx BB

cells#

1BB

cells#

1F

M

1ikiik xCPxB

1,3,5,...ifor x2

)1i(cosxC kki

2,4,6,...ifor x2

icosxC kki

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

parameters for B1(y) and three

parameters for B3(x), while magnetic

boundary conditions on the opposite walls were enforced as periodic, that is

B2(1,y) = B1(0,y) and B4(x,1) = B3(x,0).

Specified and estimated boundary conditions for

magnetic field in test case 1.

0.0 0.2 0.4 0.6 0.8 1.0y

-0.08

-0.04

0.00

0.04

0.08

Bx

/ Bo

Estim ated

Prescribed

0.0 0.2 0.4 0.6 0.8 1.0x

0.12

0.14

0.16

0.18

By

/ Bo

Estim ated

Prescribed

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

.44

-0.3

8-0

.38

-0.3

1-0

.31

-0.2

5-0

.25

-0.2

5

-0.1

9-0

.19

-0.1

9

-0.1

3-0

.13

-0. 0

6-0

.06

-0.0

6

0.0

00

.00

0.0

60

.06

0.0

6

0.1

3

0.1

3

0.1

90.19

0.1

9

0.2

5

0.25

0.3

10.31

0.3

1

0.3

8

0.3

8

0.4

40

.44

0.4

4

-0.4

4-0

.44

-0.3

8-0

.38

-0.3

8-0

.31

-0.3

1-0

.31

-0.2

5-0

.25

-0.2

5

-0.1

9-0

.19

-0.1

3-0

.13

-0.1

3

-0.0

6-0

.06

-0.0

6

0.0

00

.000

.00

0.0

60

.06

0.13

0.13

0.1

90.19

0.1

9

0.2

5

0.25

0.2

5

0.3

1

0.3

1

0.3

8

0.38

0.4

40

.44

0.4

4

Convergence history for the hybrid

optimization for test case 1.

0 20 40 60Itera tion num ber

0.00

0.02

0.04

0.06

0.08C

ost f

unct

ion

SQ PD FPG AN M

0

200

400

600

800

# of

obj

. fun

c. e

val.

LMD E

Test case 2.In this case, B1(0,y) was approximated with

only six parameters although the actual value was a discontinuous function. The value of B3(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 B2(1,y) = B1(0,y) and B4(x,1) = B3(x,0).

Specified and estimated boundary

conditions for magnetic field in test case 2.

0.0 0.2 0.4 0.6 0.8 1.0y

0.4

0.8

1.2

1.6

2.0

Bx

/ Bo

Estim ated

Prescribed

0.0 0.2 0.4 0.6 0.8 1.0x

0.0E+0

4.0E-3

8.0E-3

1.2E-2

1.6E-2

By

/ Bo

Estim ated

Prescribed

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

.38

-0.3

1

-0.3

1-0

.31

-0.2

5

-0.25-0

.25

-0.1

9

-0.1

9

-0.1

3

-0.1

3-0

.13

-0.0

6

-0.0

6-0

.06

0.0

0

0.0

00

.00

0.06

0.1

9

0.25

0.38

0.3

8

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

.25

- 0.1

9

-0.1

9-0

.19

-0.1

3

-0.1

3

-0.0

6

-0.06

-0.0

6

0.0

0

0.00

0.0

6

0.0

6

0.0

6

0.13

0.13

0.1

3

0.1

9

0.19

0.19

0.25

0.25

0.2

5

0.31

0.3

1

0.31

0.38

0.38

0.44

0.4

4

0.44

Convergence history for the hybrid optimization for test case 2.

0 20 40 60 80Itera tion num ber

0.0

0.2

0.4

0.6

0.8C

ost f

unct

ion

SQ PD FPG AN M

0

200

400

600

800

1000

# of

obj

. fun

c. e

val.

LMD E

Geometry and boundary conditions for test case 3.

g0=9.81 m/s2

Tc

Th(x)

InsulatedInsulated

x=0 x=1

B1 (y) B2 (y)

B4 (x)

B3 (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 1.0y

-2.0E-5

-1.0E-5

0.0E+0

1.0E-5

2.0E-5

Bx

/ Bo

Estim ated

Prescribed

0.0 0.2 0.4 0.6 0.8 1.0x

0.8

1.2

1.6

2.0

By

/ Bo

Estim ated

Prescribed

Specified and calculated magnetic and field distributions for test case 3.

Specified and calculated temperature field distributions for test case 3.

-0.43 -0.43-0.36 -0.36

-0.29 -0.29-0.22 -0.22

-0.16 -0.16-0.09 -0.09-0.02 -0.02

0.050.05

0.12

0.12

0.19

0.19

0.19

0.26

0.26

0.33

0.33

0.39 0.39

0.460.46

0.53

0.53

-0.43 -0.43-0.36 -0.36-0.29 -0.29

-0.22 -0.22-0.16 -0.16

-0.09 -0.09-0.02 -0.02

0.05 0.05

0.12

0.120.190.26

0.26

0.26

0.39

0.46

0.46

0.53 0.53

Convergence history for the hybrid optimization for test case 3.

0 10 20 30 40Itera tion num ber

0.0

0.2

0.4

0.6

0.8C

ost f

unct

ion

SQ PD FPG AN M

0

400

800

1200

1600

# of

obj

. fun

c. e

val.

LMD E

DISCUSSIONIn 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 L2- 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.