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thesis entitled
DEVELJPMENT AND EVALUATIW of
3 D OPTICAL sscrramw m2 THE Invasrraflnau 0F
FIBER ORIENTATION DISTRIBUTION IN INJECTION MOLDED
FIBER PEINFORLED dTHE/QMO PLASTICS

present ed by

Bin Lia/l

has been accepted towards fulfillment
of the requirements for

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degree in

 

 

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MSU Is An Afflnnotlvo Action/Equal Opportunity IMRWOHM

DEVELOPMENT AND EVALUATION OF
3-D OPTICAL SECTIONING FOR THE INVESTIGATION OF
FIBER ORIENTATION DISTRIBUTIONS IN INJECTION -MOLDED,
FIBER REINFORCED THERMOPLASTICS

By

Bin Lian

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1994

ABSTRACT

DEVELOPMENT AND EVALUATION OF
3-D OPTICAL SECTIONING FOR THE INVESTIGATION OF
FIBER ORIENTATION DISTRIBUTIONS IN INJECTION-MOLDED.
FIBER REINFORCED THERMOPLASTICS

By

Bin Lian

Calibrations and measurement accuracy estimates were carried out to evalu-
ate and improve the novel technique developed at MSU for the investigation
of the three-dimensional fiber orientation distributions for fiber reinforced
thermoplastics. Comparison of the results given by optical sectioning and sur-
face ellipse methods suggested no significant difference of these two methods.
This new method is accurate, non-destructive, and much faster than the cut
and polish method. Appropriate statistical descriptions for the second order
orientation tensor components and their corresponding errors were also devel-
oped. Error estimates were performed to identify the potential sources of
errors which contribute to the measurement uncertainty. The injection appa-
ratus is shown to produce specimens under industrially relevant conditions
and the imaging and analysis software can provide accurate and detail infor-
mation describing fiber orientation distribution rapidly. The influences of pro-

cessing conditions on the FODs were also defined.

To my wife and my parents

iii

Acknowledgments

I would like to express my deepest appreciation and regards to my
advisor Dr. John McGrath for his continued support and guidance throughout
the length of this project. Also I would like to thank Dr. Jeff Wille for the
invaluable discussions and insights. Thanks are due to several exchange stu-
dents from Germany, Armin Noethe, Jan Ladewig and Nils Schoche for their
great assistance. I must thank Dr. Gilliland of the Statistics Department for

the insightful comments about the statistical description.

I am thankful for the support of Michigan State University and the
State of Michigan Research Excellence Fund awarded through the Composite

Materials and Structure Center.

iv

Pn

tva

Nomenclatures

second order orientation components

half height of mold cavity (H/2)
Brinkman Number

fiber orientation vector

fiber diameter

Graetz Number

height of mold cavity

thermal conductivity

fiber length

length of mold cavity

refractive index

Pearson Number

time required to fill the mold

average flow velocity

flow direction coordinate (zero at gate)
transverse (in plane) coordinate (zero at vertical midplane)

optical axis coordinate (zero at horizontal midplane)

Greek
or thermal diffusivity

n power index of power law model (1]: my" —1 )

. n polymer melt viscosity

9 Eulerian angle with respect to z-axis

q) Eulerian angle in x-y plane with respect to x axis

‘1’ (6, (b) orientation distribution function

vi

Table of Contents

List of Tables ............................................................................................ i
List of Figures ......................................................................................... ii
1. Introduction ......................................................................................... 1

1.1 Injection Molding of Short Fiber Reinforced Thermoplastics. 1

1.1.1 Industrial Injection Molding Technique for Polymer Processing... 1

1.1.2 Concept of Plastic Reinforcing ................................................... 3
1.1.3 Short Fiber Reinforced Thermoplastics ....................................... 4
1.2 Mathematical Description of FODs ......................................... 5
1.3 Investigation Tools for Studies of FODs .................................. 8
1.3.1 Traditional Methods ...................................................................... 8
1.3.2 Optical Sectioning ......................................................................... 10
1.4 Optical Sectioning for SFRTP Research ................................... 10
1.4.1 Basic Concepts of Optical Sectioning ...................... i .................. 10
1.4.2 Advantages of Optical Sectioning Method .............................. 12
1.4.3 Injection Molding Apparatus ..................................................... 15

vii

1.4.4 Imaging Apparatus ...................................................................... 16

1.4.5 Image Processing Software ........................................................... 17
1.4.5.1 3-D Thinning .................................................................... 17
1.4.5.2 3D-FODAS 4 Extraction of FOD Information ............... 19
1.4.5.3 FIBOR - Ellipsoid Visualization .................................... 20

1.4.5.4 PSEUDO - Generation of Images with Specified FODs.. 21

1.4.6 Some Perspectives for the Development of the

Optical Sectioning Technique ...................................... 21

2. Calibration .............................................................................................. 23
2.1 Calibration Using Pseudo-data ................................................... 23
2.2 Single Fiber Calibration ............................................................... 24
2.3 Orthogonal Specimen ................................................................... 25

3. Comparison of FODs Determined by Optical Sectioning

and Cut/Polish Method ................................................................ 27

4. Image Processing ................................................................................. 29
4.1 Image “Crushing” to Remove Fiber Clumps ............................... 29
4.2 Influence of Non-Uniform Voxel Sizes on FODs ...................... 31

5. Error Estimates and Analyses .................................................... 33

viii

5.1 Motivation ....................................................................................... 33
5.2 Possible Sources of Errors .............................................................. 34

5.3 Confidence Limit on FODs - Standard Error of Tensor

Components ............................................................................ 37

5.4 Estimation of Measurement Error ............................................. 4O

6. Influence of Processing Parameters on FODs ...................... 43
6.1 Injection Molding Process ............................................................ 43
6.2 Processing Parameter Influences on the FODs ...................... 45

7. Results and Discussmns ......... 49
7.1 Calibrations ............................................................................. ‘ .......... 49
7.1.1 Pseudo Data Calibration ................................................................ 49
7.1.2 Results from Orthogonal Scans ...................................................... 50

7.2 Comparisons of FOD Results from Optical Sectioning and

Cut/Polish Method ............................................................... 54

7.3 Imaging Processing .......................................................................... 58
7.3.1 Fiber Clumps “Crushing” ............................................................... 58
7.3.2 Results of Influence of Non-Uniform Voxel Size ............................ 61
7.4 Estimates of Measurement Errors ............................................. 63

7.5 Results of Influence of Processing Parameters on FODs ........ 65

7.5.1 Experimental Results of Influence of Processing Parameters on

ix

FODs ..... .................................. 65

7.5.2 Results of Simulation of FOD Evolution During Mold Filling

Process .............................................................................. 68

8. Conclusions and Recommendations for Future work... 74

8.1 Calibrations ....................................................................................... 74

8.2 Image Processing ............................................................................... 75

8.3 Statistical Analysis ........................................................................... 76

8.4 Experimental Investigations of FODs .......................................... 76

8.5 Other Considerations ...................................................................... 77
Appendices... ................................................................................................ 79
A. Ratio Approach for Standard Error Calculation ......................... 79

B. Specimen Geometry ............................................................................ 80
List of References .................................................................................. 81

List of Tables

. Refractive Indices of Glass ........................................................................ 13
. Refractive Indices of Transparent Industrial Polymers and

Possible Matches with Glass ....................................................... 14
. Operating Range of Injection Molding Apparatus ................................................... l6

. Tensor Components and Unit Vectors of Major Axes of

the Ellipsoids (Tessellation Order 2) ........................................... 52
. Difference in Directions Between Individual Scans and the Mean .......................... 53
. Comparison of Tensor Components for Scans Using Different Voxel Sizes ............... 62

. Comparison of Angle Differences from the Mean Direction for

Scans Using Different Voxel Sizes ............................................................ 63
. Comparison of Experimental Uncertainty and Theoretical Estimates ........................ 64
. Error Contribution in Percentage ................................................................................ 64

xi

10.

11.

12.

13.

14.

15.

16.

17.

List of Figures

Schematic of Injection Molding Apparatus .............................................. 2
Determination of FODs by Cut/Polish Method ........................................ 9
Schematic of Optical Sectioning ............................................................. 11
A 2-D Image: From Which 3-D Image is Constructed .............................. 12
Injection Molding Apparatus .................................................................. 15
Imaging System ...................................................................................... 17
3-D Thinning ........................................................................................... 18
Visualization of FOD by 3D-FODAS ..................................................... 20
Visualization of Second Order Tensor ...................................................................... 21
Orthogonal Scans ....................................................................................... 26
Sean Locations for Optical Sectioning and Cut/Polish ............................. 27

Sample Image with Fiber Clumps and the Schematic of the Image

Preprocessing for Fiber Clump Removal ..................................... 30
Sampling Error versus Number of Tracer Fibers ..................................... 39
3 x 1 Repeated Scans in the Shell and Core layers .................................. 42
Formation of Solid Boundary Layer and Velocity Profile ........................ 44
Visualization of Result from the Four Orthogonal Scans ........................ 51
t-distribution for n=4 ................................................................................. 55

xii

18.

19.

20.

21.

22.

23.

24.

Comparison of <a11> and <a33> Given by Optical Sectioning and
Cut/Polish MethOd and the Corresponding Significance Level of
Difference (Sample n08) .......................................................................... 56

Comparison of <a11> and <a33> Given by Optical Sectioning and

Cut/Polish Method and the Corresponding Significance Level of

Difference (Sample ob7) .......................................................................... 57
Effect of the Preprocessor for Real Images with Fiber Clumps ............... 60
Influence of Temperature and Injection Speed on FODs ......................... 66
Material Effect on FODs ........................................................................... 67
Material Influence on FODs for PMMA and Nylon 6/6 ............................ 71

Evolution of FODs for Tmelt (PMMA) at 260C .......................................... 72

xiii

Chapter 1

Introduction

1.1 Injection Molding of Short Fiber Reinforced Thermoplastics

1.1.1 Industrial Injection Molding Technique for Polymer Processing.

Injection molding thermoplastic technology has a long history of more than
one century. It is gaining more attention and has acquired an increasing share
of the manufacturing industry. As one of the most common technologies in the
polymer processing industry, weight saving, remoldability, great surface
appearance and low cost of the finished parts give this technique many advan-
tages over the traditional metal casting method. Capable of high volume of
production, good product quality consistency and low labor intensity are sev-
eral other important technology niches for today’s successful manufacturing

operation. These factors make injection molding a practical choice for plastic

part production, among other manufacturing technologies. Recyclability, as
one of the major environmental concerns, also pushes people to switch to the
more environmental friendly plastic products among which a large fraction

are made by injection molding.

There has been tremendous progress in injection molding technique since the
first prototype was built. Reaction injection molding (RIM), resin transfer
molding (RTM) and many other modifications and applications are just a few
examples of technologies which grew out of the basic injection molding con-

cept. Figure 1 shows the schematic of a simple injection molding machine.

hopper

barrel heater

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Figure 1. Schematic of Injection Molding Apparatus.

Because of high viscosity, polymer melts cannot be poured into the mold.
They must be injected into the mold cavity by applying large forces with a
plunger. Moreover, once the mold is filled with melt solidification takes place

where the hot melt meets the cold wall. Therefore an additional amount of

melt must be packed into the mold to offset polymer shrinkage during cooling

process and to achieve an accurate reproduction of the mold cavity geometry.

Injection molding equipment consists of two major parts: the injection unit
and the clamping unit. The function of the former is to melt the polymer and
inject it into the mold, whereas the clamping unit holds, opens and closes the
mold automatically, and ejects the finished product. Present injection units
are almost exclusively of the in-line reciprocating screw type. The screw both
rotates and undergoes axial reciprocal motion. When it rotates, it acts like a
screw extruder, melting and pumping the polymer. When it moves axially, it
acts like an injection plunger. The screw is, in general, rotated by a hydraulic

motor and its axial motion is activated and controlled by a hydraulic system

(Tadmor and Gogos, 1979).

1.1.2 Concept of Plastic Reinforcing

Plastic is not as strong as metal on a strength per unit volume basis, but the
incorporation of fibers into pOlymer matrices increases the strength and stiff-
ness as well as the heat deflection temperature (the temperature above which
the material deforms significantly under load). Fiber reinforcement also
increases the flexural modulus to a magnitude comparable to that of tradi-

tional construction materials. Other advantages of reinforced plastics include

better performance in creep and temperature resistance, as well as reduced
thermal expansion and mold shrinkage. Among all these advantages, cost
effectiveness and its environmental friendly nature are also important rea-
sons to use fiber reinforced plastics as substitutes for metals. However, the
addition of brittle fiber materials can reduce the impact strength of thermo-

plastics (Rayson, McGrath and Collyer, 1986).

In a typical industrial manufacturing process for fiber reinforced thermoplas-
tics. the reinforcing agents which can take on the form of fibers or flakes are
mixed with matrix materials and fed into the extrusion machine, where the
mixture is heated and extruded and cut into small pellets. The pellets can be
injection or compression molded into finished products in the same manner as

traditional unfilled plastics.

1.1.3 Short Fiber Reinforced Thermoplastics

Among all the available composite fabrication techniques, chopped short fiber
reinforced thermoplastic (SFRTP) injection molding is the most feasible for
high volume production. It has the obvious advantages of excellent reproduc-
ibility and low labor intensity which means low cost. Normal injection mold-
ing machines for plastics can be used for injection molding of filled plastics.

However, sometimes it might be necessary to use special screws and barrels to

prevent excessive wear of the machinery since some glass fibers can be abra-

sive.

The outstanding mechanical properties of reinforced plastics stem from the
effective load transfer from the external stress to the reinforcing materials.
For short fiber reinforced thermoplastic composite, care must be taken to
ensure that the fiber aspect ratio must be higher than the critical aspect ratio

to retain the objective of reinforcing.

The performance of fiber reinforced thermoplastic products not only depends
on how much reinforcing material is inside the composite (volume fraction), it
is also determined by the orientation of the fibers which can produce a highly
anisotropic material. It is necessary to describe the properties in terms of vol-
ume fraction and fiber orientation distribution (FOD). For a fixed glass con-
tent, the FOD information becomes critical because the FOD will be related to
various processing parameters and to bulk properties. Based upon this infor-
mation, decisions about the processing can be made to meet the criteria of the

design objective.

1.2 Mathematical Description of FODs

After the fiber orientation distribution information has been extracted from

the measurement, we need to describe this information. One obvious choice is
the analytical or numerical distribution function, which gives a complete

description of FODs. The probability of a fiber being oriented between the
angle 61 and (61 + d6), (p1 and (It)1 + (It), and having a length between L1 and

(L1 + (114) is

P(6139$91+d6,¢1.<_¢ 91),,ng SL1+ dL) = w(e,,¢,,L,) sineldOdOdL [1]

where ”w(P,L) (dL)dP= l .
FL

The distribution function approach has been found to be workable in two
dimensional cases provided some assumptions are made to simplify the prob-
lem. It is impractical to apply it to three dimensional cases in relation to both
computer power and space needed to store the data. Advani and 'Iircker (1987)
proposed a tensor description for the fiber orientation distributions which has
been widely accepted. The orientation tensors are created by forming dyadic
products of the direction vector p and integrating over all directions, weight-
ing the product with the distribution function.
jjpipjpk. - -p,L'v (I), L) deL
atijk...l= LP [2]

J J W (p, L) dde

LP

 

where t=0 represents a number-average orientation distribution, and t=1 rep-

resents a weight-average orientation distribution. Since the distribution func-

tion has even polarity, i.e., up (p, L) = W (-p, L) , all the odd-rank tensors are

zero. This approach is analogous to using Taylor’s series (or polynomials) to
approximate a complex function. The higher order tensors provide the better
approximations. Advani and Tucker’s review (1991) showed that only the sec-
ond and fourth-rank tensors are needed for most purposes, and usually only
the second-rank tensor is used in the prediction of FODs. However a closure
law has to be incorporated into the equation to reduce the fourth order terms

to second order terms.

The second order tensor can be written as following:

 

 

 

(011) = (sinzecosztb) [3]

(012) = (sinzecososino) [4] 3’2

(a13) = (sin9cosGCOS¢) [5]

(«220 = <a12> ‘ [6]

(a22) = (sinzesinw) [7]

(a23) = (sinGcosGsin¢) [8] ’ 2' Y
(031) = (013) [9]

(“32) = (“23) [10] (flovil,d)i(rection)

(a33) = (00329) [11]

where each tensor component is the length weighted average of all the fibers

in the population, defined as the following:

2’). ' of)
(oil) = k [12]

2’1.

k

 

where [k is the length of the kth fiber and a; is the orientation contribution of

the kth fiber.

1.3 Investigation Tools for Studies of FODs

1.3.1 Traditional Methods

At present, one of the most common methods to obtain three dimensional fiber
orientation data involves cutting and polishing the sample down to the loca-
tions where FOD are needed. Analysis of the shapes of the elliptical fiber

cross sections exposed by physical sectioning is performed as shown in Figure
2. The in-plane orientation angle o is determined from the direction of the

major axis of the exposed ellipse, while the out-of-plane angle 6 is calculated

from the ratio of the major to minor axis.

 

 

 

 

 

Figure 2. Determination of FODs by Cut/Polish Method.

This method falls into the category of metallographic polishing technique. The
disadvantages of this technique are obvious since it is destructive, time con-
suming, and might introduce ambiguity into the orientation data because
fibers with angles 9 and It - 9 have the same cross sections. Elaborate effort is
needed to eliminate this ambiguity using techniques such as optical diffrac-
tion or physical etching. The other shortcoming of this method is that it does
not provide fiber length information, which is very important for several rea-
sons. For example, fiber-fiber interaction modeling is closely related to the
fiber length, the mechanical properties also depend strongly on the length dis-
tribution, and the length information can also affect the confidence level calcu-

lated for the FOD determinations.

10

1.3.2 Optical Sectioning

An unique technique has been developed in our laboratory at Michigan State
University for the investigation of three dimensional fiber orientation distri-
butions (Wille, 1993). It is nondestructive, capable of providing detailed and
complete 3-D information, as well as being accurate and rapid. This technique
utilizes optical sectioning, which will be explained in detail in the next sec-

tion.

1.4 Optical Sectioning for SFRTP Research
1.4.1 Basic Concepts of Optical Sectioning

In our studies, optical imaging is made possible by matching the refractive
indices of the matrix material (PMMA) and the glass fiber (BK-10). For most
short fiber reinforced thermoplastic composites, the bulk materials usually
are opaque or translucent because either one of the components of the compos-
ite is opaque, or the interfaces between the fiber and matrix strongly scatter
incident light due to mismatched refractive indices. By matching the refrac-
tive indices of PMMA and BK-IO (n=1.49), we have been able to produce

transparent composite samples consistently. A small amount of opaque tracer

ll

fibers (O.2Wt%) is introduced into the mixture to provide statistical represen-
tation of the behavior of the glass fibers. An inverted light microscope is used
to obtain the image of the tracer fibers. The optical system was modified by
introducing a condenser and diffuser to obtain a very narrow depth of field
which is approximately the same as the fiber diameter (15 microns). This con-
figuration proved to have minimum information loss and required minimal
storage space for the data. In this optical sectioning method only the portion of
the sample which intersects the focal plane within the thickness of the depth
of field of the optical system is visible. This is what is referred to as optical

sectioning. The complete 3-D representation of the fiber orientation in the

 

 

 

 

 

'1'
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1’
I
I
I
1’
I
I
I
I
I
I

I

I

I
I

I

I

.. scan (i+1)
scan (i)
scan (i- l )

 

   

 

I

:trans arent composi
,(PM + glass fi
I

 

 

Figure 3. Schematic of Optical Sectioning.

12

scan cell is obtained by stacking the 2-D images sequentially, as shown in Fig-
ure 3. A picture of an actual 2-D image is given in Figure 4. The black segment
is the portion of the tracer fiber which intersects the focal plane. The negative
of these images are compressed and stored since they require less disk storage

space.

 

Figure 4. A 2-D Image: From Which 3-D Image is Constructed.

1.4.2 Advantages of the Optical Sectioning Method

Compared with the traditional cut/polish method, the optical sectioning

method has proved to be accurate, nondestructive, and rapid. The cut/polish

13

technique may introduce ambiguity into the fiber orientation data, and
requires correction to account for differential probability of fibers with differ-
ent orientation and length intersecting the polished plane. For the optical sec-
tioning method, the material selections are limited since the specimen have to
be transparent. Nevertheless, we have a reasonably wide variety of polymer
materials to select from with different refractive indices. The refractive index
of the glass can be modified by controlling the addition of chemical com-
pounds, making it possible to apply this technique to other material system
that one may be interested in. Table 1 and Table 2 list the refractive indices of

most common glass and industrial polymer materials.

Table l: Refractive Indices of Glass.

 

 

 

 

 

 

 

 

 

 

 

Glass - Refractive index Remarks
BK-IO glass 1.49
Quartz 1.55
E-Glass 1.56
C-Glass 1.57
D-Glass 1.59-1.63 depends on composition

 

14

Table 2: Refractive Indices of Transparent Industrial Polymers and Possible
Matches with Glass.

 

  

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Material Refractive Remarks
Index
Poly(teuafluoroethylene) 1.35-1.38
Poly(vinyl fluoride) 1.46
Poly(butyl acetate) 1.463
Poly(ethyl acrylate) 1.468
Poly(methyl acrylate) 1.48
Poly(mcthyl methacrylate) 1.49 BK- 10
Polypropylene 1 .49-1 .50 BK- 10
Cellulose acetate 1.50
Polyethylene (Low Density) 1.51 depends on degree
Polyethylene (High Density) 1.53 0f ”Stanini‘y
Polyisoprene(Nature Rubber) 1.52
Polyacrylonitrile 1.52
Polybutadiene 1.52
Nylon 11 1.52
Nylon 6:6 1.53
Acrylonitrile-butadiene-styrene 1 .538
Poly(vinyl chloride) 1.54
Styrene acrylonitrile copolymer 1.57 061355
Epoxy Resins 1.57-1.61 depends on C-Glass
composrtron
Polycarbonate 1 .58
Poly(ethylene terephthalate) 158-1 .63 D-Glass
Polystyrene 1.590 D-Glass
Polyimide 1.70

 

 

 

 

 

 

15

1.4.3 Injection Molding Apparatus

VVrthout buying a costly industrial injection molding machine but still retain-
ing the characteristics ’of the real industrial process, a laboratory scale injec-
tion molding apparatus prototype was built (Wille, 1993). It features a simple
ram type injector. The sandwich construction of the mold design allows flexi-
bility in changing the geometry of the mold cavity at very low cost. Tempera-
ture and velocity measurements are accurate and easily set. Figure 5 is a the

schematic of the injection molding apparatus setup.

      
   
 
     

Hydraulic Speed Regulation
Cylinder
Hydraulic

Position
Transducer

 

 

Injection
Machine

  

 

 

 

 

Acquisition

 

Figure 5. Injection Molding Apparatus.

16

The operating ranges of processing parameters achieved with this system are

relevant to real industrial processing and are listed in Table 3.

Table 3: Operating Range of Injection Molding Apparatus.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameters Range
Melt temperature 20°C 3 Tmelt 3 300°C
Mold temperature 20°C 3 T 3 120°C
mold
In'ection speed , 3 3
J 1005'1 S‘yS 18005—1;2.5cm /s<v< 30cm /s

Injection/hold pressure 32, 000173 i

Mold and gate geometry Flexible in Mold and Gate Design

Fiber length 0.1 mm S length S 3mm

Fiber contents 0 < c < 30 Wt%

Materials Other Matrix/Fiber Combinations Possible

(see Table 1 and 2)

1.4.4 Imaging Apparatus

The optical imaging system includes a Nikon DIAPHOT inverted light-micro-

scope, three programmed servo-motors for automated three-dimensional scan-

ning of specimens, video-camera and frame-grabber for image acquisition and

digitizing, and an Amiga-3000 multitasking computer for data processing and

storage. The schematic is showed in Figure 6. Since the optical scanning sys-

17

tem is automated by the computer, it takes much less time to get the same

amount of information compared with the cut/polish method (Wille, 1993).

 

 

 

 

 

 

 

 

 

   
 

Motion Controller
Board

..........

....................................

Figure 6. Imaging System.

1.4.5 Image Processing Software

1.4.5.1 3-D Thinning

A three dimensional thinning program is used to process the raw image for

extraction of fiber orientation distribution information (Wille, 1993). Fiber

18

materials in the raw images is represented as voxels where each fiber is about
several voxels in diameter and tens or hundreds of voxels in length. To deal

with the complerdty of these structures and to reduce computation time, an

 

raw image

 

thinned image

Figure 7. 3-D Thinning.

19

image skeletonizing algorithm was implemented and used to reduced the raw
images to thinned images. A fiber in the raw image is reduced to a thin line
segment from which the orientation and length of the original fiber is deter-
mined. After thinning each voxel has less than two nearest neighbors, i.e. the
connectivity is preserved while providing much easier determination of orien-
tation information from these thinned images. Figure 7 illustrates the effect of

3-D thinning for the three dimensional image.

1.4.5.2 3D-FODAS - Extraction of FODs Information

After the image is thinned, it needs to be analyzed and the fiber orientation
distributions within the'image must be described in one way or another. In
our studies, two methods have been used for the FODs description. A second
order as well as a fourth order tensor description are given since the orienta-
tion and length information for each individual fiber are unambiguously
known. We also use a direct way to represent the FODs visually in the form of
a three-dimensional spherical histogram (Vlfrlle, 1993). A subdivided icosahe-
dron (a sphere divided equally into 20 triangular facets) is used to represent
the possible orientation of fibers. How many times the triangular facets are
subdivided is referred to as the tessellation order. The elevation above the

“sea-level” of the sphere represents the accumulation of fiber materials

(length) that coincide with the direction (0, (p) of the corresponding triangu-

20

lar facets, as illustrated in Figure 8.

 

Figure 8. Visualization of FOD by 3D-FODAS.

1.4.5.3 FIBOR - Ellipsoid Visualization

A tensor visualization program is also available which creates graphic repre-
sentation by projecting the ellipsoid of the second order tensor t0 the x-y, y-z,
and x-z plane (Schoche,1993). An example is shown in Figure 9 for a second
order tensor given as follows:

'0700 0.000 0.100

(01,-) = 0.000 0.015 0.000
0.100 0.000 0.015

21

 

 

 

   

y- ’ z 0
L. i... Ly

Figure 9. Visualization of Second Order Tensor.

 

 

 

 

 

 

 

 

 

1.4.5.4 PSEUDO - Generation of Images with Specified FODs.

After the image acquisition system had been implemented, a software
(PSEUDO) was also developed to provide noise-free images to test how well
the image processing program work. The software is capable of producing a
simulated image which contains cylindrical fibers that have well defined
shapes, lengths and orientations. Fibers with specified length and diameter

can be produced.

1.4.6 Some Perspectives for the Development of
Optical Sectioning Technique

Having finished setting up the system and many major problems concerning

the workability of the optical sectioning technique having been solved, exper-

22

iments are needed to answer the following questions concerning the useful-

ness and accuracy of this new measurement tool:

1) Are the imaging system and the image processing software capable of pro-
ducing accurate and consistent results?

2) Are the FOD results produced by the optical sectioning comparable to data
in the literature which is given by the traditional cut/polish method?

3) What are the effects of changing the parameters in the image processing
software such as voxel sizes, Tessellation orders? Are their effects significant?
4) Among all the possible sources of errors for the FOD measurement, which
are those which are the most significant and important?

5) To achieve the objective of FOD research for SFRTP design, what are the

most important processing parameters that influence the FODs?

The above questions has been carefully considered throughout this project and
positive answers and explanations have been acquired and will be shown in

the following chapters.

 

Chapter 2

Calibration

2.1 Calibration Using Pseudodata

A software program called PSEUDO has been developed to create simulated
images which contents fibers of desired and known orientation. Since the
image is noise free, it can be used to quantify the accuracy of the analysis soft-

ware and detect any bias (systematic error) introduced by the software.

An ideal 3-D random fiber orientation distribution which contained 81
straight fibers 600 um long and 10 um in diameter was created by the
PSEUDO Program using the actual voxel sizes (1.993 x 3.367 x 14.900 um)
used for real images. The pseudo fibers were placed in random locations
within the sample volume of 1077 x 797 x 1000 um and oriented uniformly in

all directions in 3-D space. For a random fiber orientation distribution, the

diagonal elements of the second order tensor are 1/3 and all off-diagonal ele-

23

24

ments are zero. The simulated image was analyzed and compared with these

known values.

2.2 Single Fiber Calibration

Previous tests (Wille and Schoche, 1993) demonstrated that the optical imag-
ing system and the analysis software can measure the fiber length and angle
for individual fibers to very high accuracy (2° error for in-plane and out-0f-
plane angle measurements and 3% error in length measurement) compared
with other available techniques (approximate 10° error in angle measurement

and no length information).

To define the uncertainty in our measurements, tests were carried out by mea-
suring the in-plane angles of two carbon fibers and comparing them with the
angles measured by the analysis software (Wille, 1993). These tests were
repeated by r0tating the sample for a total of 90 degrees with 10 degree incre-
ments. A maximum error of 2 degrees in fiber angle and 3% in length were
found, which are the measurement uncertainty for angle and length for in-

plane fibers.

Tests were also done to estimate the uncertainty in measuring out-of-plane

25

angles. A carbon fiber was placed in the air tilting out of the horizontal plane
and was measured using the optical imaging system, it was found that the
maximum error for the out-of-plane angles is about 2 degrees and the length

measurement was less than 2% in error (Wille, 1993).

2.3 Orthogonal Specimen

Since there is not a standard fiber orientation distribution available to be com-
pared with, the other calibration experiment was carried out by cutting a
small volume 0f dimension 10.3mm x 1.5mm x 1.5mm from the center of one of
the samples. A specified volume (1.077mm x 0.797mm x 0.797mm) within this
specimen were scanned in four orthogonal directions sequentially. The second,
third and fourth scans were made by rotating the specimen 90°, 180°,270°
with respect to the first scan direction, as shown in Figure 10. The four images
were thinned, analyzed and rotated mathematically back to the same viewing
(reference) direction as the first scan. If certain kinds of systematic errors
were present, for example, which could possibly be created by non-uniform

voxel sizes, the FODs from these four scans will not be the same.

26

 

WT Flow

 

 

Direction

Figure 10. Orthogonal Scans.

Chapter 3

Comparison of FODs by Optical Sectioning

and Cut/Polish Method

To verify that the optical sectioning technique provides results that are com-
parable to the results given by the traditional method, samples were investi-
gated using optical scanning and then subjected to physical cutting and
polishing followed by surface ellipse analysis of the same volumes. The data
were taken from the middle of the samples (x/L ~ 0.5) as shown schematically

in Figure 11 and comparisons were made for two samples, sample n08 and
y
x [
z
. . >3x1 Scans
| m
x =—'—'

Figure 11. Sean Locations for Optical Sectioning and Cut/Polish.

8
”U
S.
to
S

 

 

 

i
w

9
a.
co
5
co
3

 

 

 

 

 

27

28

sample 0b7.

3x1 scans were made to obtain a single measurement. A 1x I scan cell corre-

sponds to the minimum imaging area in x-y plane defined by the optical sys-
tem which is 1077 x 797 um and the z dimension (150 um) is obtained by
putting together 10 slices of 2D images 15 pm thick each. Comparisons were

made at four different through-thickness (2) locations and twelve transverse

(y) locations.

In this comparison experiment, optical sectioning was carried out before the
samples were polished down to various through-thickness locations. The

shapes of the ellipses on the cutting plane were analyzed. The rotation angles
0 were calculated by measuring the angle between the major axis of the
ellipse and the x-axis (flow direction). The elevation angles 0 were obtained by

comparing the ratio of the major axis a to the minor axis b of the ellipses:

 

 

 

 
   
 

¢= mac—:31) us]
2 2 y I
0: acos[~r(y4-y3)2+ (x4-x3)2] [14] /
(yZ-yl) + (x2—x1) (x434) (x1,y1)
4 x

 

Chapter 4

Image Processing

4.1 Image “Crushing” to Remove Fiber Clumps

For given sharp and clean fiber images, the 3D thinning software works well
to reduce the raw images to thin line segments. However, sometime the image
quality is not so good and contains non fiber information, such as shadows cre-
ated by abnormal fiber aggregation, dirt and voids present in the specimens,
these are referred to as “fiber clumps”. The shadows are most likely to occur
for images taken deep inside the specimens since in this region the optical
path is partially blocked by the opaque fibers sitting in between the focal
plane and the optical lens. The thinning program processed the fiber clumps
down to hairy structures which significantly influenced the FOD results, espe-
cially for the core layer. The magnitude of <a 11> was reduced slightly, but the
magnitude of <a33> was increased from the typical value of 0.1~0.2 to 0.3~0.4.

Based on the original program written by Jeff Wille, an image preprocessing

29

30

algorithm was implemented to deal with the problem. This preprocessor
divides the whole image plane into small square cells of dimension about 50 x
50 microns and checks the percentage of the voxels set in each cell. If it is over
40%, a square is defined which has the same center but twice the size of the

original cell, voxels inside it are erased, as showed in Figure 12. The dimen-

 

before after

Figure 12. Sample Image with Fiber Clumps and the Schematic of the Image

Preprocessing for Fiber Clumps Removal.

31

sion of the cells was chosen in such a way that it was comparable to the size of
the fiber clumps but considerably larger than the fiber diameter. The 40% cri-
teria takes into account the possibility that two fibers could be in the same cell
even for good images and these should not be erased. The possibility of having

more than three fibers inside a same cell is so low that it is negligible.

To verify that the algorithm‘ does not affect good quality images but does
remove the fiber clumps if they exist, two test cases were studied. A clean
image containing a random fiber distribution and real images obtained by

through-thickness scanning were both used for the tests.

The random fiber orientation distribution created by the PSEUDO program
were used to check how much the algorithm would affect the FOD results of a
clean image. Actual images obtained by optical sectioning which contained 3 x
3 scans from the surface to the midplane at x/L=0.5 were used to test the

effects of the algorithm to images containing fiber clumps.

4.2 Influence of Non-Uniform Voxel Sizes on FODs.

A possible source of systematic error involves the utilization of different voxel
sizes in the x, y, and 2 directions to represent fiber materials. The three

dimensional images are obtained by linking two dimensional images together

 

32

sequentially. The third dimension is added knowing the scanning step size in
the sectioning (z) direction. The voxel sizes in the x, y, and z direction are
3.367, 1.993, and 14.900 um respectively for the optical imaging system. It is
of interest to determine whether the nonuniformity in voxel size will affect the
quantitative results for FODs significantly, noting that even though the thin-
ning takes place uniformly in all directions, the non-uniform voxel sizes could

possibly create bias FOD results.

Analysis was done using the sample for the orthogonal scans and two step
sizes were used in the sectioning direction to obtain two different voxel sizes,

i.e., 14.900 and 7.450um.

 

Chapter 5

Error Estimates and Analyses

5.1 Motivation

Error estimates and analyses are very critical in this study in order to prove
that the optical sectioning method is accurate while possessing other advan-
tages over traditional techniques. Any measurement tool should not only have
the capability to produce experimental data, but also should provide confi-
dence limits for the information obtained from it. Uncertainty estimate analy-
sis is important relative to the improvement of system accuracy by identifying
the sources of errors and the sensitivity of overall uncertainty to various
parameters. It is also important to define the confidence level for those data
that have been produced such that quantitative comparison of FODs of the
SFRTP fabricated under different processing conditions can be made. Thus

conclusions with quantitative confidence limits can be derived.

33

34

5.2 Possible Sources of Errors

Errors and uncertainties in fiber orientation measurement can be classified
into three categories: systematic, measurement and sampling errors. System~
atic error is any nonrandom difference between measured and actual quanti-
ties which can not be eliminated by using larger sample sizes or more
repetitions. Measurement error is related to the random variation expected
when a measurement is repeated on the same sample and which can not be
improved by using larger sample sizes. Sampling error denotes the variation
among measurements expected for different samples from the same popula-

tion.

fixatemfiumrs:

This kind of error stems from the inaccuracy or bias of the measurement tools
and methods which creates persistent differences between the measured val-
ues and the actual values. These kinds of errors can only be eliminated by
using better measurement equipment. They may be identified by comparing
measurement results using other techniques. In our study, the following fac-
tors might be most important:

- alignment of sample on the optical stage

comments: several degrees of systematic error can be introduced

35

into the angle measurement if it isn’t done correctly.
Make sure sample is free of flash and attached to the
stage correctly.
- calibration of the optical system including alignment, magnification,
etc.
comments: check occasionally and make sure there are no loose
parts in the optical path.
- possible bias created by imaging system and analysis software
comments: the thinning program could shorten the fiber length by
the amount of one fiber diameter, which doesn’t notice-
ably influence the results. The non-uniform voxel sizes
could also create bias FOD results, tests done showed no

significant influence.

Wm
Uncertainty in the quantities being measured, condition of the measure-
ment tool, operator effect and other factors all contribute to the measurement
errors. Care must be taken to minimize the possible errors.
- resolution limit of the image digitizing
comments: this imposes an uncertainty of angle measurement of

about 4.3 degree (based on Ad(z—resolution) = 14.9um,

L = ZOOum, M _=_ 43° ). This is comparable to the
Alxn

36

discrete angle interval used to describe FODs (for tessel-
lation order 4, the interval is 5 degree).
- change and fluctuation of light intensity

comments: light intensity has a lot to do with the FOD results
since it not only changes the apparent fiber length but it
also influences the results related to the through-thick-
ness direction because the image of fibers tends to stay
on the screen for a longer time if light intensity is higher.
It should be high enough to make sure that all of the
individual fibers are connected to get the correct length
information but low enough so as not to exaggerate the z
dimension. Careful adjustment of light intensity is the
key point to reduce this error to a minimum, and a better
power supply or stabilizer could be used to eliminate the

intensity fluctuation.

Solarium

Sampling error describes the inherent variation within the measured
quantity or quantities. Measurements should be made repeatedly for samples
produced under the same condition so that the measurement can be treated as
random independent quantities. In the investigation of fiber orientation dis-
tribution, it means we need to measure the orientation of a single fiber from

each individual sample of a large enough population. It is nearly impossible to

37

0 produce and collect FOD information from such a large sample size. To deal
with this problem, we usually assume that fibers which are about one fiber
length apart can be considered as independent and sampling can be done
within these locally homogeneous regions. This assumption is a true approxi-

mation if fiber-fiber interaction is a short range phenomenon.

5.3 Confidence Limit on FODs

- Standard Error of Tensor Components

It has been widely accepted that processing influences the FODs in SFRTP.
The confidence limits on the FODs are as important as the FODs themselves,
especially when comparisons are to be made to determined if different pro-

cessing conditions produce different FOD results.

Fiber orientation distributions are approximated by second order tensors
which’consist of nine tensor components, within which only five of these nine
components are independent. The geometrical analogy of the second order
tensor is an ellipsoid. These five parameters are needed to give a full descrip-

tion of the ellipsoid, which includes the shape (aspect ratio) and orientation.

From the definition of the tensor components, for example, for <311>3

38

N o
I
2 [i ' “11
(all) = (sin20cos2¢)= EJfi—- ’ [15]
i: I
(where I, , ail are the length, the tensor component for the ith fiber and N is
the total number of fibers). It can be seen that it is a weighted average of all
the fibers in the sampling population. We would like to know how accurate
these parameters are for certain fiber orientation distributions. Noting that

<all> is one of the five parameters (tensor components) to describe the FODs,
the standard deviation of the mean <311> is an appropriate measure of the
uncertainty of <all>. Since <all> is a length-weighted average of all the all

for each individual fiber within the scan cell, the standard deviation of the

mean (standard error) which gives the accuracy of <all> at 95% confidence

level can be calculated as follows (Advani and Tucker, 1992):

 

/ N 2) f N I
211' (“ii (011))2211'2
i=1 ,

i=1 _2.0-SD(011)

£1, [21,2]- ‘m

\ i=1 / i=1

515mm): 2.0

 

 

 

[16]

 

 

 

 

where SD(a11) is the standard deviation of (an) and N is the total number of
fibers sampled. For a given homogeneous p0pulation, SD(a11) describe the dis-
persion of the fiber orientation and will be the same for any subdomains (par-

titions) within the total population, provided these subdomains have enough

fibers to give statistical meanings. An empirical relation can be derived

39

between the magnitude of SE(an) and number of fibers N if the range of

SD(an) is available. Figure 13 demonstrates typical ranges for SD(a11) for the

shell and core layers, as well as the SE(a11) ~ N relation deduced from them.

 

Sampling Error
o o
H i—A
c or
I I

.O

O

U!
l

 

0.00

N

/

 

l

\\

%

1

Core Layer

Shell Layer

    
  

”ll/l“

l 1 l

\
all».

 

 

100

200

Number of Tracer Fibers (N)

300

Core Layer:
0.25 < SD(311) < 0.35

Shell Layer:
0.15 < SD(an) < 0.25

Figure 13. Sampling Error versus Number of Tracer Fibers.

As we will see in the next section, the measurement errors (usually about 0.01

to 0.02) is consistently smaller than the sampling errors (standard errors, 0.02

~ 0.13). The biggest error, i.e., the standard error will be reported in this

study, except when pointed out specifically. It is also believed that the system-

atic error is smaller than the measurement errors since no significant biased

result has been detected.

40

5.4 Estimates of Measurement Error

Comparison between the experimental uncertainty for the second order tensor
components and the estimates of the error analysis is necessary to determine
which measurement errors are most important to the overall uncertainty so
that they can be identified and improved. An uncertainty estimate for the
measurement errors was performed. Having acquired an explicit expression
for the tensor components in terms of the measurement variables (angle,
length), a sensitivity study was carried out to determine which measurement
uncertainty contributes the most to the total variation. An estimate of the
total uncertainty is also defined. For a given relation:

Af(xpx2,-~,x)=~[(g——f1)(Ax)2+(g——f22)(mz) + n+(gijzmmz [17]

n

 

 

The following gives the calculation for the estimated measurement error for

tensor component <all>:

From the definition of <all>:

21,.- (sin0)2- (cost)2
<a,,>= " [181

2’.-
i

 

the sensitivity coefficients can be found as follows:

41

_1_ . 2. 2
‘a—<a >= (sin0)2- (cos¢)2- ._1_ ... 1 E (5109) (COSO)
ali ll 1

(21): 24

 

[19]

I

2
21,.- sin0icosOi (0050,.)
5535““): [20]

21.-

 

 

 

2
3 _ 21‘.- (sin0i) sinoicosoi
Wam- 21‘ [211
andfinally,
A(a11)= «[2183] —(all)) (Al) 2+2(.a—ae— (011)) (A0 +Z(—a-(all))(Ao
[22]

3 x 1 scans (each 1077 x 797 x 150 um) were repeated ten times for both core
and shell layers respectively, as illustrated in Figure 14. The means and
uncertainty of the tensor components were calculated. The error estimates
were performed knowing the orientation and length information of the fibers

in these images.

3x1 scans (shell layer)
2 ;‘
t. ..

 

 

 

 

 

 

3x1 scans (core layer)

 

Figure 14. 3 x 1 Repeated Scans in the Shell and Core Layers.

Chapter 6

Influence of Processing Parameters on FODs

6.1 Injection Molding Process

Injection molding is a very complicated process. Basically it includes: (1) the
melting of the filled or unfilled polymeric material; (2) mold filling during
which the pressurized melts flow into the mold cavity; (3) the packing and (4)
cooling. Theories of fluid mechanics, polymer rheology, and heater transfer are

all needed to explore the physical nature of this process.

Substantial efforts have been put into the investigation of the mold filling pro-
cess for short fiber reinforced thermoplastics, both in theory and experiment
by different research groups. Several commercial versions of simulation soft-
ware are already available. Tucker (1991) and Wang (1993) give very nice

reviews about these work in this area.

43

44

For example, during the filling process of a simple mold geometry such as a
rectangular slit, heat transfer occurs from the hot melt to the cold mold
through the cavity wall and produces a through-thickness nonuniform tem-
perature distribution. Under this non-isothermal condition the temperature
field during injection molding is characterized by thin cold boundary layers
near the walls and a hot core region away from the walls. The significance of
the influence of the solid boundary layers on the flow field and eventually the
FODs depends on the ratio of the solid layer thickness and the height of the
cavity. The effect of the solidified layers upon the through-thickness velocity

profile is shown in Figure 15:

cavity wall

    

gate flow direction

Figure 15. Formation of Solid Boundary Layer and Velocity Profile.

The heat transfer process can be characterized by several non-dimensional

numbers (Tucker 1991):

2 . .
G 2- 4b _ heat conduction tzme

_ mm]- moldfilling time ;(Graetz Number)

 

 

11V2 _ heat dissipation rate . (Brinkman Number)

Br = — ,
k (Tmu — Tm”) heat conduction rate ’

 

 

45

Temp. diflerence between mold and fluid
: T —— T = , P N b l
P" C ( ”'9“ m”) Temp. diflerence to change viscosity ( earson um er

The thickness of the solid boundary layer is approximately given as

-—[-)—= ottfm , which indicates that a high injection rate produces thin solid

J52

boundary layers and a slow injection rate leads to a thick boundary layer. Due
to the highly viscous nature of polymer melts there is significant local energy
build-up created by shear heating and if this energy is not dissipated fast
enough through conduction or convection, a temperature increase will occur
which will change the viscosity of the melt and eventually the velocity profile.
This effect can not be neglected since the Brinkman number is around 10 for
a typical injection molding process. The Pearson number is always large (~10)
(for §~ l/ 20°C ), indicating that a solid boundary layer is constantly formed

on the mold wall.

6.2 Processing Parameter Influences on the FODs

One of the characteristics of fiber reinforced, injection molded parts is that the
fiber orientation distribution displays different behaviors at different loca-
tions through the thickness. This variation has been called the layered struc-
ture. Fibers near the cavity wall experience high shearing and tend to be

aligned along the flow direction which creates a highly aligned shell layer.

46

Fibers near the centerline of the cavity experience no shear and carry the ini-
tial orientation state (assumed to be random) near the gate into the mold cav-

ity, leading to the somewhat random core region.

There are a number of processing parameters which could influence the FODs
, however they are not equally important. To identify the most significant ones
is a very important process as it is linked directly to optimal design for short
fiber reinforced thermoplastic parts. The process variables are listed below

and their relative importance are also discussed.

I . |° S I I 1 fl | I
Since the differential equations for the second order tensor components
contain only the first derivative with respect to length and time, this
means that the FODs depend solely on the total (shear or extension)
deformation but not on the rate of deformation (shear or extension
rate). Thus FODs produced by different injection speeds will be the
same under isothermal conditions. However, low injection speed leads
to a thick solid boundary layer near the wall which creates a con-
strained flow path favoring higher fiber alignment along the flow direc-
tion and producing a thick shell layer and a thin core layer. This
phenomenon is most important when the thickness of the solid bound-
ary layer is not negligible compared with the thickness of the cavity.

O

Another effect that will make the results deviate from the isothermal

47

prediction is the shear thinning effect which relates to the fact that
melt viscosity is shear rate dependent, a common feature observed for

most polymer materials.

Wm

The influence of the melt temperature on FODs mainly comes from the

temperature dependence of the power law index n (0<n<1, n: my” I ).
For Newtonian fluids n equals one. The lower the melt temperature, the
closer n approaches zero, i.e., the non-Newtonian fluid behavior, which
corresponds to a blunt velocity profile and a thick core layer. For real
industrial processes, the operating temperature range is limited to
about 50 degrees for most polymer materials since high melt tempera-
ture tends to degrade the material and low temperature requires high
processing pressure. The change of n in this temperature range is small
and it usually won’t produce significant changes of the FODs. For
example, It varies only from 0.19 to 0.27 for a melt temperature change

of 40 degrees (PMMA).

Wm

Usually the mold temperature is kept low to reduce cycle time in actual
industrial processes. For'most materials the mold temperature is much
lower than the polymer glass transition temperature and a solid bound-

ary layer will be formed. This phenomenon is characterized by a large

48

Pearson number (~10) in the injection molding process (Bay and
Tucker, 1991). Changing the mold temperature usually has little effect

on the velocity profile and the FODs.

W

It was found that FODs were quite different for samples made using the same
kind of materials but different grades (molecular weight). For PMMA of low
molecular weight (Dupont Acrylic 2010) which would be expected to exhibit
more Newtonian-like behavior, the shell layer was found to be thicker than
those samples made using PMMA of higher molecular weight (Kodak). Simu-
lations were performed to explore the difference in FOD results from samples

produced using different PMMA.

Chapter 7

Results and Discussions

7 .1 Calibrations
7.1.1 Pseudo Data Calibration
A random fiber distribution created by PSEUDO was processed using the

image processing software. The result given by the analysis software is

showed as below:

0.3373 -0.0014 —0.0005 0.3333 0.0000 0.0000
[ail-l = -0.0014 0.3365 -0.0012 . [aij] = 0.0000 0.3333 0.0000
—0.0005 —0.0012 0.3265 0.0000 0.0000 0.3333

The ideal random distribution has a magnitude of 1/3 for the diagonal ele-
ments and zero for all off-diagonal elements for the second order tensor. The

actual result showed that the error introduced by the imaging and analysis

49

50

was less than 2% for the second order tensor components and the average

measured length was 576 um which was a 4% error.

7.1.2 Results from Orthogonal Scans

Bias in the FOD results may be introduced due to the fact that the voxel used
to represent the digitized image has different dimensions in the x, y and 2
directions. If the imaging system and the image processing software are accu-
rate, the four images obtained from the orthogonal scans and the fiber orienta-
tion information extracted from them should be exactly the same. Figure 16
demonstrates the result from these scans. Note that while the details of the
FODs are not identical in each view, the basic shape and orientation are simi-

lar.

51

I‘IIbv-IuI-d

I: I'm a. u- <

 
    
 

a
?

   
   

V
3335‘
"IZJ

E

41.
'1."

«1‘: '
£23
A». 58.74
V

47¢?
v

:3“?
’3:

n%’

e».
4'?

A

        
   

G
\
‘1'
n

>
.V
\

 

m"--. I“ ...... ......

fies
, 1"“:1‘, J
.«I {33%; Err
' mum = i

:.- VAW" ‘V'

   

    
    
   

       

Y ’2 1'! ‘
may
‘74s}?

Figure 16. Visualization of Result from the Four Orthogonal Scans.

Table 4 shows the 2nd order tensor components for these four scans (after
being rotated back to the same reference orientation) as well as the compo-
nents of the unit vectors emajor corresponding to the directions of the major

axes of the ellipsoids which represent the most populated fiber direction.

52

Table 4: Tensor Components and Unit Vectors of Major Axes of the Ellipsoids.

 

 

 

 

 

 

 

 

 

 

 

 

(Tessellation Order 2)
aij emajor

0.4730 0.1402 0.2800 0.7253 '

“a" 1 0.1985 -0.1662 0.3505
sym 0.3284 0.5926
“0.4370 0.0787 0.3062 0.6910

scan 2 0.1514 0.1321 0.2352
sym 0.4116 0.6835
0.4243 0.1423 0.2697 0.6801

5“" 3 0.2210 0.1705 0.3828
sym 0.3547 0.6253
0.4096 0.1222 0.2536 0.6420

scan 4 0.1767 0.1794 0.3526
sym 0.4137 0.6809

 

 

 

 

 

 

 

 

If these four scans are very similar, the directions of the major axes should
coincide very closely with each other. The angles between any two of the four

unit vectors and the difference from the average direction are summarized in

Table 5.

53

Table 5: Difference in Directions Between Individual Scans and from the Mean.

A. A.
1
AU: acos( 6 major - e'major)

 

 

 

 

 

 

 

 

 

 

 

 

Angles
(in degrees)
A A scan 1 scan2 scan3 scan4

AU: aCOS(elmajor ' Elmajor)
scan] 8.64 3.62 6.9]
scan2 9.11 7.27
scan3 4.16
scan4

__ '7 ——*

At: acos(e major - emajor) 5.82 7.27 5.33 5.43

e‘major represents the direction of the major axis from the individual scan and

=

e is the average direction. A if is the angle difference of the major axes in

major
degrees between any two of the four scans, and K. is the angle difference of

individual scan and the average. It can be seen from the above results that the
difference between the individual measurements and the average is approxi-
mately 5 to 7 degrees, which agreed with the uncertainty in angle measure-

ment (section 5.2).

Analysis also showed that using higher tessellation order (>2) did not signifi-
cantly affect the accuracy of the tensor components. Nevertheless, higher tes-

sellation order should be used to obtained accurate length information for

54

individual fibers since the higher the tessellation order (angular resolution),
the less the chance that several fibers will be put into the same “bin” of the

subdivided icosahedron.

7.2 Comparisons of FOD Results Obtained from the Optical Section-
ing and Cut/Polish Method

Figure 18 and Figure 19 are comparisons of the FOD results determined by
the two methods within which the lateral (cross flow direction) FOD varia-

tions for different through-thickness locations are shown.

Each data point is the average of results from three adjacent scan cells
obtained by the 3 x 1 scans in the flow (x) direction. The error bar represents

the variation (standard deviation) of <all> or <a33> within these three cells.

Also presented in Figure 18 and 19 is the significance level of differences .
between the tensor components obtained from the two methods. The statisti-
cal significance (p value) of the difference between tensor components was cal-
culated from the mean and standard deviation of the measurements. The t

statistic of the difference between the means Was calculated using

 

 

55

2 ' 2
2_ (n1—1)51+(n2-1)52

 

 

 

_ [24]
P (n1—1)+(n2—1)
The t-distribution can be calculated as:
1 n+1
I.(n+ ) 2 _ 2
F(:)=J' 2 (1+6) ) dx [25]

“Mr

{3
2
where the n is the number of degrees of freedom n=n1+n2-2.

The typical shape of the t-distribution is illustrated as follow:

‘ F(t)

- — — — — — — if , ...... '. .'.'.'.--..- -

    

 

Figure 17. t-Distribution for n=4.

The p-statistic (0 < p <1) is the shaded area in Figure 17 and it represents the
possibility of observing the difference of the means from the two populations
greater than the difference of the two means of the known distributions, if the
experiments were repeated under the same conditions. The magnitude of p
can be affected from the two populations being compared basically in two
ways: first, the difference of the means, second, the spread of the distribu-
tions. The more the means deviate from each other, or the smaller the stan-

dard deviation of the distributions, the smaller p will be, the more distinct of

 

 

 

 

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Figure 18. Comparison of an and a33 Given by Optical Sectioning and Cut/

Polish Method and the Corresponding Significance Level of Difference (Sam-

ple n08).

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Figure 19. Comparison of an and a33 Given by Optical Sectioning and Cut/

Polish Method and the Corresponding Significance Level of Difference (Sam-

ple ob7).

58

the difference will be for the two distributions.

The above comparison suggests that the optical sectioning method is capable
of producing information similar to FODs obtained by the surface ellipse
method. The significant level p suggested that even though results given by
these two methods didn’t agree for every points, the overall picture indicated
there was no significant difference. The experiment also showed that the opti-

cal sectioning is about five times faster than the traditional cut/polish method.

7.3 Imaging Processing

7.3.1 Fiber Clumps “Crushing”

An image processing algorithm was implemented to remove fiber clumps
present in the actual images. Tests were performed to define how well this
algorithm works and the results showed that it didn’t affect clean images (free
of fiber clumps) but did remove the clumps when they were present (results
shown below). This approach will shorten the fibers near the clumps, but due
to the fact that there are only a few clumps in the images, the total extent of
shortening of real fiber lengths in the sampled population is insignificant. The
following is a comparison of tensor components with and without the prepro-

cessor for a clean image created by the Pseudo program and it shows the ten-

59

sor components are changed by less than 2%.

0.3373 0.0014 0.0005 0.3384 0.0015 0.0014
laijl= 0.0014 0.3365 0.0012; laijl= 0.0015 0.3391 0.0002
0.0005 0.0012 0.3265 0.0014 0.0002 0.3225

(a) W (b) W

The test using actual images showed that for poor images the effect of correct-
ing for fiber clumps is obvious for images with fiber clumps, as shown in Fig-
ure 20, which shows reduced <a33> component and slightly increased <311>
component. The preprocessing did not affect the FOD results at all where the
image quality was good, for example the shell layer, because it is here where
the through—thickness scans start and images are usually sharp and clean.
However, when scannings go deeper down toward the core region, fiber clumps

are likely to occur, it was found that <a33> in the core layer could be reduced

down by approximately 50% by the image preprocessor even though the

change in <all> is not that significant. It is obvious that how much change

caused by the image preprocessing to the second order tensor components

depends on'the image quality.

61.)

Effect of Clump Crushing on FODs

 

 

 

 

l 1 1 1 r 1 1 fl 1 1
0.9- q
08“ ................... ...l.’ i..-..—.-- ..~~“ .1
0.7” If“, .. _
x7
80-6_ ..... ‘ ....... I . .. ..
m : ,/ : . i__Crushed
'0 . / . . , . .
gO-SP ................ ': / 2 . ....... .1
z: / : j--Uncr'ushed
cu ’ ; : ;
0.4... ..,. . ..................... ..
/ .
I 3 “a11
/ .
0.3- .......................... .............................................. ..
-__ : i xxa33
02,. ......‘.\.\.‘% a
O 1 1 1 1 1 1 1.- 1 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z/b

Figure 20. Effect of the Preprocessor for Real Images with Fiber Clumps.

The above results (“crushed”) were also compared with FODs obtained at the
same scanning locations for the same sample, but scanning was started from
the middle by polishing one side of the sample to the centerline (z/b ~ 0). By
scanning this way, the image quality is better for the core layer where fiber
clumps are most likely to occur if scans is made from the sample surface (z/b ~

1). The comparison showed that the magnitudes of <all> and <a33> agreed

with each other (Figure 18 and Table 8). The reason why the image prepro-

61

cessing algorithm affects the <a33> more than the <all> probably is due to the
fact that the contribution of the clumps to <311> is small compared with the

total contributions of all fibers since most fibers are already aligned in the flow

direction, but the clumps affect <a33> significantly because they create appar-

ent fiber components with a strong z-component and they play a strong role in

determining the magnitude of <a33>.

It is concluded that the preprocessor is necessary since results with a high
confidence level require a larger amount of tracer fibers, which is more likely
to create fiber clumps. The algorithm can also help to get rid of non-fiber

image content such as dirt, voids, etc.

7 .3.2 Results of Influence of Non-Uniform Voxel Sizes

Two different voxel sizes in the optical sectioning direction (VoxSizeZ) were

tested for the orthogonal sample, i.e., voxel sizes 3.367 x 1.993 x 14.900 and
3.367 x 1.993 x 7.450 um were used. Table 6 contains the orientation tensors

as well as the unit vectors corresponding to the major axes of the ellipsoids.

62

Table 6: Comparison of Tensor Components for Scans Using Different Voxel Sizes.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

' ‘ VoxSizc.Z=14.900 11m VoxSize.Z=7.450 11m

aij emajor 311 8major

0.0890 0.2642 0.7088 0.4738 0.0969 0.2812 0.7303

0.1418 0.1210 0.2429 0.1543 0.1331 0.2593

0.4042 0.6623 sym 0.3719 0.6319

0.4054 0.1170 0.2860 0.6574 0.3780 0.0990 0.2362 0.6656

0.1952 0.1642 0.3334 0.2746 0.1235 0.2161

0.3993 0.6757 sym 0.3473 0.121 1

0.4073 0.1388 0.2217 0.6681 0.4112 0.1455 0.2526 0.6619

0.2349 0.1558 0.4089 0.2547 0.1866 0.4337

‘ 0.3577 0.1276 sym 0.3341 0.6114

0.3992 0.0843 0.2660 0.6799 0.3707 0.0981 0.2360 0.6672
0.2241 0.1213 0.2927 0.2814 0.1374 0.2189 Ii
0.3767 0.6723 sym 0.3479 0.1139 ]J

 

 

 

 

 

 

 

Table 7 shows the variations of directions of these unit vectors from the aver-

age as well as the mean of the variations for the two different voexel sizes,

which changes from 5.00 degrees to 4.26 degrees. Theoretically, if uniform

voxel sizes are used, for example, 2 x 2 x 2 um ,to represent fibers, there will

not be any bias for the three orthogonal directions. The results indicated that

changing the voxel size toward uniformity doesn’t change the results signifi-

cantly, but the image scanning time and storage space were doubled.

63

Table 7: Comparison of Angle Differences from the Mean Direction for Scans Using
Different Voxel Sizes.

 

 

 

 

 

Ki: acos( eimajor . emajor) scan] scan2 scan3 scan4 A: I (At) 2/4
VoxclSizc.Z=14.900 4.78 2.06 8.35 2.00 5.00
VOXCISiZC.Z=7.450 6.93 0.97 4.19 2.49 4.26

 

 

 

 

 

 

 

 

The above results lead to the conclusion that the present choice of voxel sizes
has no significant influence on the accuracy of the FOD results but requires

less storage space for the images.

7.4 Estimates of Measurement Errors

Table 8. compares the results from the experiment and the theoretical error

estimates for both the shell and the core layer, where d<au> and d<a33> are

the standard deviations of the ten measurements. For the experiment, 3 x 1
scans were repeated ten times and the mean and standard deviation were cal-
culated. For the error estimate, the sensitivity coefficients were calculated
using orientation and length information for each fiber using the equations

derived (Eq. [22], section 5.4). Estimates of measurement errors were 5

degrees for both angles (0, 0) and 15 pm for the length 1. It can be seen that

the results obtained from experimental and theoretical estimates are consis-

64

tent in magnitude. This confirms that the measurement errors come mainly

from the angles and the length.

Table 8: Comparison of Experimental Uncertainty and Theoretical Error

 

 

 

 

 

 

 

 

 

Estimates.
Location Results <a1 1> d<a1 1> <a33> d<a33>

shell layer Experimental 0.866 0.009 0.020 0.007
Uncertainty
Theoretical 0.012 0.003
Error Estimate

core layer Experimental 0.440 0.017 0.1 1 1 0.020
Uncertainty
Theoretical 0.014 0.009
Error Estimate

 

 

 

 

 

 

 

 

The contributions to the measurement errors due to uncertainty of length and
angle are tabulated in Table 9. It can be seen that the in-plane x-y angle 0
contributes the most to the measurement uncertainty for both shell and core

layers. The length contribution becomes significant for the core layer.

Table 9: Error Contribution in Percentage.

 

shell layer 0.000064 (13.6%) 0.000006 (16.3%) 0.000080 (70.1%)

core layer 0.000026 (42.7%) 0.000031 (4.0%) 0.000133 (53.3%)

2(§E(a11>)2(az)2 2(3%(a11))2(750)2 2(-a%;(all))2(A¢)2

 

 

 

 

 

 

 

65

7.5 Results of Influence of Processing Parameters on FODs.

7 .5.1 Experimental Results of Influence of Processing Parameters on
FODs.

Samples were made under different processing conditions and comparisons of
FODs are shown in Figure 21 (also see Thesis, Ladewig, 1993). The processing
conditions for each samples are also shown in the same Figure. The first

graph compares <311> and <a33> of two samples produced with different melt

temperatures while all other processing conditions were maintained the same.

No significant differences in FODs were detected except for <all> near the

center. The other set of samples were made with different injection rates but
all other conditions were the same. The FODs of the sample with the low
injection rate showed a thicker shell layer, which could be explained by the
formation of a solid boundary layer near the wall. For low injection speed, the
solid boundary layer has more time to develop due to the heat transfer from
the hot melt to the cold wall of the mold and a constrained flow channel is
formed which favors high fiber alignment along the flow path and create a

thicker shell layer. This observation agrees with the theoretical expectation.

 

 

 

 

 

 

 

 

 

1.0
1 ----- ..... ai1-DP7
0.87
m , TMe11=2°° °C
(3 0'6‘ —°'— a11-DP9
,. 0.4: TMen=185 C
E .
”7"” 33-DP7
0.2: a
1 —+— 333-DP9
0.0‘ ' 1 ,
-1 0 1
z/b
Fiber Orientation of Specimen DP? and DP9
Melt Temperature Influence
1.0
‘ °°°° °"" a11-DP2
0.8“ - . ..
.c ' V =8.6 chs
cc j »‘ I ' Piston [ l
('0 ‘. .'
g 0'6, '- If" —°— a11-DP9
g . .- . -' V
d 1 . . '. ' =2.5 CW5
a 0'4‘ 15.-i . Piston [ ]
0.2~ "/ - —-°-~ ass-0P2
"/ —’—a33-DP9
0.0 - . ~ “‘
-1 0 1

ll!)

Fiber Orientation oi Specimen DP2 and DP9
Injection Speed Influence; Pn=2.9

 

Figure 21. Influence of 'Ibmperature and Injection Speed on FODs.

FODs for samples produced using two different grades of PMMA were also
compared as in Figure 22. PMMA produced by Eastman Kodak has higher '

molecular weight and viscosity, while PMMA provided by Dupont has lower

67

molecular weight and lower viscosity. It shows that the FODs of the sample
made with high molecular weight has a thicker core layer which may be
caused by the highly non-Newtonian behavi0r of the material where the shear
thinning effect is significant. The solid curves are the through~thickness <311>
and <a33> components of 3 x 3 scans taken at x/L ~ 0.5 for sample produced
with low viscosity PMMA and the error bars represent the standard deviation

of the means (<all> and <a33>). The dotted lines are the average of FODs of

Influence of Materials on FODs

 

1 l l l T l l l l l
' ’ _ 06wv1scbsny100P0NT PMMA) ' '
0.9... .................. ........ ......... ......... .................... § —1
I , ..... High ViSCOSlty(KC)DAK PMMA) ‘ :3: I
03,. .......................... . .......... .......... ' .......... ' ................................. —1
i Q1— 3 .
0.7,. ' ............................... . ....... ‘ ..... _
a: ; 2 ;
(006,. ...................................... ._ I ,—
N l : . _~ .
13 C I j 1
$05.. ................. I ....................... . .1
C504__ I .................. [I ........... .1

O
(D

l
l

 

 

 

0.2-. ...... I .................................. d
: f In e : - : '
01.. ....... i ......... 5 , ........ ..... _ ......... ' ...... .IgwI .......... 1 ....... ...
‘ ' ' ' l ‘ 3"'==. i
I” 1 1 1 1 1 1 "L“
-1 —08 -06 -04 -02 0 02 04 06 08 1
z/b

Figure 22. Material effect on FODs.

68

four samples made with high viscosity PMMA under the same conditions, the
error bar represents the variation of the second order tensor components from

sample to sample which shows good reproducibility of the results.

7.5.2 Results of Simulation of FOD Evolution During Mold Filling
Process

To understand the effects of material properties on the FODs, a simulation
using the finite difference method was used to predict the evolution of FODs

during the mold filling process.

J effery’s equation is usually used to describe the motion of dilute suspensions
in Newtonian fluids (Jeffery, 1923). Theoretical and experimental studies
showed that non-Newtonian fluids (Gauthier, Goldsmith and Mason, 1971),
non-linear flow fields (Shanker, Gillespie, and Guceri, 1991) have a measur-
able but relative small effect on the fiber orientation. The interaction between
fibers appears to be the most significant “non-Jeffery” effect in these problems.
Thus in this studies, J effery’s equation was applied to describe the motion of
fibers and Tucker’s interaction model was used to modeled fiber-fiber interac-

tion (Tucker, 1987).

The distribution function approach was used instead of working directly with

69

the second order tensor. The purpose was to acquire useful information about
how the orientation of individual fibers evolve during the mold-filling process.
A combination of Jeffery’s equation and the Tucker interaction model (Tucker,
1987) yields the following equation of motion (rotation) for a single fiber

within the flow field:

 

, 6 av av 2vya av
6= 2 — sinGcosB— — (sin6)-— + (0056)2 + sinOcosO—y
re +1 3x a? By

 

l . av, zavx a_:y dvy
—|: 2 ] — smecosoa + (c056) 8_y - (sin6)2a + sin6cose—
re +1

 

3y
frag
W86
[26]
' avx rZ—l . ¢ . e e [27]
=— n 5 cs
([1 6y ,3.” 31 cos¢ 1n 0

where 9 is the angle of the fiber projection onto the x-z plane makes with the

x-axis ,and the velocity gradient is in the z direction, while (11 is the angle the

fiber makes with the y-axis. CI is the interaction coefficient and re the equiva-

lent ellipsoidal axis ratio. The distribution function is ‘i’.
3. 2

GD
l"
.<

1, X
(flow direction)

70

Since the viscosity of polymer material depends on the shear rate and temper-
ature, they will also influence the flow field (velocity profile) of the mold-filling
process. The dependence of viscosity upon shear rate and temperature can be

described by a power law:

n(i,T)= n,(T>1"" 1281

where Y is the magnitude of shear rate and n is the power law index. Differ-
ent materials or same materials with different grades usually have quite dif-

ferent power law indices, and temperature can have moderate effect as well

(Tadmor and Gogos, 1979).

Two test cases were studied for different power low indices n: n=0. 19 (PMMA)

avx

)l

and n=0.65 (Nylon6/6) with the same shear rate (7 = ~ 1000/s ) near the

 

 

wall. Random fiber orientation distribution was assumed for the initial state,
and the time scale and a geometry similar to the operating conditions of the
injection apparatus for this study were used. Figure 23 shows the evolution of

<all> through thickness along the flow direction for these two processes. The
magnitude of <311> near wall (lz/b I ~1) increase quickly because the high
shearing near the wall aligns the fibers in the flow direction, and fiber-fiber
interaction disturb this alignment to some degree so that <all> stays. some
where below 1.0 which corresponding to perfect alignment in the flow direc-
tion. The orientation distribution in the core region (z/b ~0) doesn’t change

much since fibers experience no shearing in this region. It can be seen that

71

Evolution of a11 in Injection Molding

 

-1 0 M (flow direction)

z/b (thickness direction)

n=0.19 (PMMA)

Evolution of a11 in Injection Molding

  

-1 0 xlI (flow direction)

2!!) (thickness direction)

n=0.65 (Nylon 6/6)

Figure 23. Material influence on FODs for PMMA and Nylon 6/6 (at T=220C).

72

PMMA (n=0.19) displays the influence of significant non-Newtonian behavior.
The PMMA simulation shows a thicker core layer which agrees with our
experimental finding (section 7.5.1), and the simulation results for Nylon6/6 is

similar to the data from Tucker’s group (Bay and Tucker, 1991).

Another comparison was made to explore the effect of the melt temperature.
The melt temperature of PMMA was increased from 220C to 260C and the
power law index changes from n=0.19 to n=0.27 (Tadmor and Gogos, 1979). No
significant difference was observed for FODs for melt temperature 220C and

260C, as shown in Figure 24, which agreed with experimental observations

(section 7.5.1).

Evolution of a11 in Injection Molding

O

 

-1 0 x/I (flow direction)
z/b (thickness direction)

n=0.27

Figure 24. Evolution of FODs for Tmelt (PMMA) at 260C.

73

The above results showed the influence of the effect of shearing on the fiber
orientation distributions for isothermal processes. The effect of the injection
speed is not shown here. Since the derivatives with respect to time and spatial
coordinates in the equations of motion for the fibers are all first order, it indi-
cates that the FODs will be the same for sample produced using different
injection speed to a first order approximation (isothermal) (Tucker, 1984). If
the heat transfer can not be neglected and isothermal condition is not satis-

fied, the injection speed will have some effect on FODs (see section 7.5.1).

Sophisticated software should be used if heat transfer, extensional flow and
other factors such as complicated geometry of mold are needed for specific

cases.

Chapter 8

Conclusions and Recommendations

for Future Work

8.1 Calibrations

Single fiber calibrations verified that the optical sectioning method can accu-
rately measure the orientation (2 degrees error) and length (3% error) of indi-

vidual fiber.

In the calibration using the random fiber orientation distribution created by
PSEUDO, the FOD results given by the image analysis software suggested
that the software can provide accurate and unbiased fiber orientation infor-
mation even though non-uniform voxel sizes were used, the resulting FOD is
very close to the known values of the second order tensor components
(<a11>=<a22>=<a33>=1/3, <312>=<a13>=<a23>=0) with less than 2% differ-

ence .

74

75

The calibration using the orthogonal sample also confirmed that the optical
sectioning can provide accurate and consistent FOD results regardless of the
scanning directions within 5 degrees for angle measurement and 3% error for

length measurement.

8.2 Imaging Processing

The experiment which utilized different voxel sizes demonstrated that non-
uniform voxel sizes didn’t significantly affect the FODs. The present choice of
voxel size proved to be feasible in terms of accuracy and storage space consid-

eration.

The incorporation of the image preprocessor is capable of removing fiber
clumps and leaving good images intact, and. the image processing time is
reduced by about 30%. It could possibly shorten the fiber length for fibers that
are close to the clumps but it is not significant. A more robust algoritth for
the thinning program might be worthwhile to be developed to add more

sophistication to deal with this problem.

76

8.3 Statistical Analysis

Since the measurement errors of the optical sectioning system is relatively
small (0.01 ~ 0.02) compared to the magnitude of the standard deviation of the
mean (0.03 ~ 0.15) for the second order tensor components, it is appropriate to

use the standard error to represent the uncertainty of’ the mean values <aij>.

On of the major objectives of this project is to relate the processing parameters
to the fiber orientation distributions. It is necessary to develop statistical cri-
teria or measures for the comparison of FODs for samples produced under
various conditions. Parametric or other approaches can be used for the com-

parisons.

8.4 Experimental Investigations of FODs.

From the FOD results acquired experimentally, it was found that melt tem-
perature did not affect the FODs significantly for the temperature range
tested, but the injection speed showed some influence on the FODs. High
injection speed tended to produced thicker core layer and thinner shell layer.

The simulation results also supported these observations.

For the future work, the effects of mold geometry (converging, diverging) will

77

be an interesting problem to be studied.

The other way to further strengthen the plastic parts and reduce warpage is
to use reinforcing ribs. The behavior of FODs within these ribs is a very
important issue to achieve the goal of reinforcing. How the layout (orientation
relative to the flow), sizing (size of ribs relative to the fiber length) of these

ribs affect the FODs is virtually unknown experimentally.

8.5 Other Considerations

One important factor that can make the optical sectioning technique more
valuable is to develop other fiber/matrix combinations which utilize matrix

materials that are popular for plastic reinforcing, such as polycarbonate.

This unique technique can also be applied to a wide variety of other applica-
tions. One interesting problem could be the thermal. stress analysis. Fibers
are incorporated into the plastics to provide enhancement of the mechanical
properties such as strength and stiffness. They can also improve the thermal
stability, such as reducing shrinkage and warpage. The FOD information is
closely related to how much improvement can be obtained. The speed of opti-
cal sectioning allows fast comparison of FODs versus product performance

and the nondestructive nature of the technique also allows other tests such as

78

mechanical tests to be carried out.

Our contact with Dr. Gilliland of the Statistics Department at MSU stimu-
lated a discussion about the other way to calculate the standard errors for the
tensor components. It will be interesting to study if there is any discrepancy
between the results computed using current method and the method proposed

by him for various conditions (sample size, distributions, etc.).

79

 

Appendices
A. Ratio Approach for Standard Error Calculation
ELI-a,“ 2 y j k
(all) = J = J ,whereyj= Lja“;
2‘7 2L1
1 J

a—11 = = ratio—of—average;

I 2 2 2
W011) = —3[Sy+<ai1>5L_2<011>5yL];
NZ
2 1 - 2 I 2 _2 .
5,» = HEW-y) = mm- ‘Nr" ]
1 J
1 2 1 2 2 .
5L - HEW-L) .. A-/—_—I[;Lj—NL],
J

l I ., ,
SyL = m2 (Yj-S’) (Lj-L) = m[ZYij—Nll] ,
J 1
Stande Error (standard deviation of the mean):

SE(all) = 2.0 /V(a“) (with 95% confidence level).

80

Specimen Geometry

 

6.

 

 

 

 

l-I

 

 

 

 

 

 

 

 

 

 

 

0‘ —— D
6.9 9 00
_ - r .................... l _
Gate *
‘1
on
Film Gated Strip

 

All Dimension in mm

 

 

 

81

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