ELASTIC PROPERTY PREDICTION OF LONG FIBER COMPOSITES USING …

COMPOSITES USING A UNIFORM MESH FINITE ELEMENT
METHOD
University of Missouri
In Partial Fulfillment
Master of Science
AUGUST 2008
The undersigned, appointed by the Dean of the Graduate School, have
examined the thesis entitled
COMPOSITES USING A UNIFORM MESH FINITE ELEMENT
METHOD
a candidate for the degree of Master of Science
and hereby certify that in their opinion it is worthy of acceptance.
Advisor: Dr. Douglas E. Smith
Readers: Dr. P. Frank Pai
Dr. V.S. Gopalaratnam
ACKNOWLEDGEMENTS
First, I would like that thank my advisor Dr. Douglas E Smith for his guidance
throughout my research at the University of Missouri. Also, a big thanks goes to Dr.
David Jack for his help with the micromechanical models as well as all of the help in
the lab over the last two years. Elijah Caselman, another former lab mate, is also owed
a big thanks for developing much of the finite element method used for this research.
Next, I thank my committee members Dr. P. Frank Pai and Dr. V.S. Gopalaratnam
for their advice and review of my research. Finally, I must acknowledge the National
Science Foundation grant DMI-0522694 for funding this research and Vlastamil Kunc
from Oak Ridge National Laboratory for providing the CT scans of a composite part.
ii
MESH FINITE ELEMENT METHOD Joseph Ervin Middleton
Dr. Douglas E. Smith, Thesis Supervisor
ABSTRACT
fiber reinforced composite materials. Calculations are performed with a finite element
mesh composed of a uniform array of three dimensional finite elements. To better
represent spatial inhomogeneities, an increased number of Gauss points are employed
during elemental stiffness calculations. Rather than making a complex finite element
mesh with element boundaries at all fiber-matrix interfaces, Gauss points are used to
define materials points as in fiber depending on where the point lies within the model.
A correction factor is applied to accommodate strain differences within elements that
contain both fiber and matrix, allowing for greater accuracy in the uniform mesh
elements. In this approach, fewer elements can be used to model a composite system
enabling computational time and demands of memory to be reduced. The method
also avoids complex meshing routines that are required when fiber-matrix interface
geometries are needed.
The uniform mesh finite element method developed here is used to predict effec-
tive properties for curved long fiber composites and for an actual long fiber composite
sample provided by Oak Ridge National Laboratory. The results of the curved fiber
uniform finite element models are compared to modified Halpin-Tsai and Tandon-
Weng micromechanical models where a good agreement is demonstrated. A con-
vergence analysis is performed to show stability and convergence of results. Image
processing techniques are applied to the Oak Ridge National Laboratory sample in
order to extract a model from the CT data which was provided.
iii
2.1.1 Halpin-Tsai Model . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Mori-Tanaka Model . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.2.1 One Dimensional . . . . . . . . . . . . . . . . . . . . 20
3.1.2.2 Three Dimensional . . . . . . . . . . . . . . . . . . . 22
3.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 23
iv
3.3.1 Spacial Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.3 Thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 Effective Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Computed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1 Split Uniform Mesh Finite Element Model . . . . . . . . . . . . . . . 63
5.1.1 Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.2 Computed Results . . . . . . . . . . . . . . . . . . . . . . . . 82
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.1 Halpin-Tsai parameters used for short fiber calculations [1] . . . . . . 25
4.1 Elastic properties of matrix and fiber . . . . . . . . . . . . . . . . . . 51
4.2 Effective property results Halpin-Tsai and Tandon Weng models with
comparison to straight fiber (curvature=0) finite element model. . . . 51
4.3 Short fiber-based results using Halpin-Tsai and Tandon Weng models
with comparison to finite element model with a curvature of 1. . . . . 53
5.1 Finite element results for 1st layer in Z-direction of the 3 x 3 x 3 model.
(Labels above indicate x, y, and z cube positions respectively, i.e. 111
starts at the origin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Finite element results for 2nd layer in Z-direction of the 3 x 3 x 3 model.
is the center cube) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Finite element results for 3rd layer in Z-direction of the 3 x 3 x 3 model.
is the cube farthest from the origin) . . . . . . . . . . . . . . . . . . . 72
5.4 Gauss point convergence for model-111 . . . . . . . . . . . . . . . . . 75
5.5 Finite Element results of complete model with 6 G.P.’s and 3x3x3
model averaged. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
vi
5.6 Finite Element results of complete model with 6 G.P.’s and Halpin-
Tsai micromechanical model assuming a distribution function of fibers
from Equation 5.2 where the parameter n is varied. . . . . . . . . . . 83
5.7 Finite Element results of complete model with 6 G.P.’s and Tandon-
Weng micromechanical model assuming a distribution function of fibers
from Equation 5.2 where the parameter n is varied. . . . . . . . . . . 84
vii
3.2 Algorithm for a 3x3 median filter [2]. . . . . . . . . . . . . . . . . . . 32
3.3 Simple cropping of an image. . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Simple threshold of an image of rice (Source: MATLAB, R2007b, The
MathWorks, Natick, MA). . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Panda image example illustrating importance of structural element
size [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7 Example of noise removal by erosion and dilation [3]. . . . . . . . . . 37
3.8 Example of erosion and dilation shown at the pixel level. (Source:
http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip-Contents.html) 39
4.1 Model of Long Fiber with curvature of 0. . . . . . . . . . . . . . . . . 41
4.5 Mesh used for all simple curved long fiber models. . . . . . . . . . . . 43
4.6 Elastic modulus property predictions for all curved fiber models. . . . 55
4.7 Shear modulus property predictions for all curved fiber models. . . . 56
4.8 Poisson’s ratio predictions for all curved fiber models. . . . . . . . . . 56
4.9 Von Mises stress plot of simple fiber model with curvature of 1 during
εxx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
εyy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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εzz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
εxy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
εyz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.14 Stress plot of simple fiber model with curvature of 1 during εzx. . . . 58
4.15 Stress plot with deformation of simple fiber model with curvature of 1
during εxx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
during εyy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
during εxy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 3x3x3 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 3D Model of a region the size of one of the smaller models. . . . . . . 64
5.4 Slice 1 of 1200 provided by ORNL. . . . . . . . . . . . . . . . . . . . 65
5.5 Showing the entire image processing procedure. a) after cropping b)
after thresholding c) after morphological smoothing d) after final rota-
tion and cropping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6 An example showing the progression of morphological operations to
remove noise from thresheld image 0001. a) original image, b) 1st
erosion c) 2nd erosion d) 1st dilation, e) 2nd dilation, f)last dilation
and final data image used for analysis. . . . . . . . . . . . . . . . . . 67
5.7 Collection of image data into column vector. . . . . . . . . . . . . . . 68
5.8 Mesh used for all 3x3x3 split models. . . . . . . . . . . . . . . . . . . 69
ix
5.9 A representation of the relative distribution of E11 within the complete
model. 3 Layers of results are shown as 3 different bar charts stacked on
top of each other. Each chart represents 9 individual model’s properties
for the Z1, Z2, and Z3 layer. . . . . . . . . . . . . . . . . . . . . . . . 73
5.10 Gauss point convergence analysis for model-111. . . . . . . . . . . . . 74
5.11 Histogram of E11/Em for the 27 split model analyses. . . . . . . . . . 76
5.14 Histogram of G12/Em for the 27 split model analyses. . . . . . . . . . 78
5.17 Mesh used for all 3x3x3 split models. . . . . . . . . . . . . . . . . . . 81
5.18 U2 displacement contour of model-111’s positive and negative X faces
during εxx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.19 U2 displacement contour of model-111’s positive and negative Y faces
during εxx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.20 U2 displacement contour of model-111’s positive and negative Y faces
during εxx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.21 Von Mises stress contours from Z-direction strain as well as correspond-
ing image 696. a)Result from complete model F.E. model, b)Result
from cube 333 F.E. model which corresponds as shown in image,
c)Image 696 showing fiber slice at level of display for both F.E. re-
sults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
INTRODUCTION
Composite materials are a macroscopic combination of two or more distinct materials
separated by a recognizable interface. Composites are widely used for their favorable
strength and stiffness to weight ratios and enjoy numerous structural, electrical, tri-
bological, and environmental applications. For this thesis, structural properties will
be investigated. A practical definition then, for composites, is a material constructed
of a continuous matrix constituent that binds together and gives form to an array of
stronger, stiffer reinforcement constituents. The reinforcement constituents investi-
gated in this research are long, cylindrical fibers in a polymer matrix. The resulting
composite material has a balance of material properties making it more useful than
either material alone.
Classification of composites occurs on two levels. The first is with respect to the
matrix material. A few examples of these are organic-matrix composites (OCMs),
metal-matrix composites (MMCs), and ceramic-matrix composites (CMCs). Organic-
matrix composites include two popular types of composites: polymer-matrix compos-
ites and carbon-matrix composites. The second level of classification refers to the form
of the reinforcement constituent. These include, but are not limited to: particulate
reinforcements, whisker reinforcements, continuous fiber laminated composites, and
woven composites which include braided or knitted fibers. The long fibers evaluated
within this thesis are classified as whisker reinforcements.
Manufacturing processes used to form long fiber polymer composites result in
1
fibers of varying length and complex disorganized fiber arrangements. As a result,
reliable yet cost effective methods for determining homogenized elastic properties of
composite materials are important to the composites industry. Determining the ef-
fective properties of discontinuous long fiber composites is the focus of this research.
No specific materials will be investigated, but rather a method of determining com-
posite material properties which is made of a general matrix material and general
fiber material. Materials will be considered to be linear elastic with no investigation
into the mechanics of the interface between matrix and fiber.
Micromechanical models were found to be less effective for long fibers. More
research is needed in long fiber micromechanical models in order to more accurately
predict long fiber material properties. The uniform mesh finite element method was
successful in predicting material properties for long fiber composites. The uniform
mesh finite element method was employed in evaluating a set of regularly arranged
curved fiber models as well as a real composite attained from CT scan data.
1.1 Organization of Thesis
Chapter One provides an introduction to composite materials and motivates the re-
search. A background of prediction methods is reviewed in Chapter Two with a focus
on analytical and numerical property prediction methods. The evaluation methods
employed in this research is presented in Chapter Three which includes an overview
of the finite element method, analytical property prediction, and image processing.
A simple set of curved long fiber models are used in Chapter Four to validate the
uniform mesh finite element method and show convergence of results. Chapter Five
2
demonstrates the computational method with a material sample which has been pro-
vided by Oak Ridge National Laboratory in the form of micro-computed tomography
image slices. Because of the large model size, the domain of structural analyisis is split
equally by three in all directions leaving twenty-seven individual models to analyze
and compare. Also, the entire model is evaluated within the constraints of computa-
tion power and memory and compared to the smaller, more refined models. Model
periodicity is implemented through a representative volume element approach. Lastly,
Chapter Six gives conclusions and recommendations for the research presented.
3
CHAPTER 2
LITERATURE REVIEW
Over the last half century, composite materials have seen large increases in devel-
opment in response to their unique and advantageous mechanical properties. The
increased application of composites has required a new method for predicting their
elastic mechanical properties. Analytical methods [4–7] using simplified methods and
assumptions have been widely used due in part to their simplicity and ease of use.
Unfortunately, the accuracy of simple analytical methods have been shown to decrease
with volume fraction and have yet to be demonstrated on long fiber composites. Un-
derlying assumptions of shape, size, and materials all idealize the composite in order
to achieve a relatively accurate solution. The two analytical methods investigated
below are the Halpin-Tsai and Mori-Tanaka models. It should be noted that material
models in this thesis were developed with a focus on short fiber composites, and will
be applied to longer fibers in subsequent sections, where possible.
Also, within the last two decades, computational costs in computing have signif-
icantly decreased. This has given rise to numeric methods for solving the problem
of material property prediction in the fiber composites. By discretizing the domain
of the model, these methods can solve the problem without some of the assumptions
required in the analytical evaluations. The finite element method and the boundary
element method for computing homogenized elastic properties will be discussed.
4
2.1 Analytical Property Prediction
One main goal in creating analytical property prediction methods is to provide poly-
mer processing communities with a measure for determining elastic properties for
injection molded short and long fiber composites. It has been shown that commonly
employed analytical methods are based on similar simplifying assumptions which in-
clude [8]:
1.) All constituents are linear elastic and isotropic or transversely isotropic mate-
rials.
2.) Fibers are cylindrical (and thus axisymmetric), identical in size, and can be
characterized by their aspect ratio (`/d).
3.) A perfect bond exists between fiber and matrix.
It should be noted that the micromechanical models presented below, Halpin-Tsai
and Mori-Tanaka, are representative of ideal materials without defects. For real com-
posites, other factors such as variability in fiber size or in constituent properties must
be included for more accuracy when predicting elastic properties in actual products.
Despite this, these simplified models have been shown to provide reasonable predic-
tion of material properties of polymer composites. Below, the Halpin-Tsai equations
and Mori-Tanaka model will be reviewed.
2.1.1 Halpin-Tsai Model
The first micromechanical model reviewed here is known as the Halpin-Tsai model.
The Halpin-Tsai equations are very simple to the useful forms of Hill’s [9] generalized
self-consistent model with engineering approximations to make them more suitable
5
for designing composite materials. Hill assumed the entire composite model to be a
cylinder consisting of continuous and perfectly aligned fibers [9]. Statistical homo-
geneity and traverse isotropy were the only requirements for the cross-section and
spacial arrangement of the fibers.
By combining the self-consistent approach of Hill with the solutions of Hermans
[10] and making a few additional assumptions, Halpin and Tsai provide a simpler
analytical form for predicting material properties [5].The Halpin-Tsai equations need
only one equation to find all the composite moduli and the longitudinal Poisson’s
ratio is simply found from the rule of mixtures. It should be noted that the Halpin-
Tsai equations are partially empirical. One parameter, ζ, was found by fitting the
equations to numerical results.
The Halpin-Tsai equations provide reasonable results for E11 at low aspect ratios,
but is less accurate at higher ratios [8]. It also gives better results for E11 and G12
at low volume fractions and decreases in accuracy as the volume fraction increases.
Hewitt and Malherbe [11] suggested that the parameter ζ should be a function of
volume fraction. This was done to improve results for G12 by fitting a new equation
to finite element results of a two dimensional problem . Also, Lewis and Nielson
improved upon the Halpin-Tsai equations at high volume fractions [12]. By adding
a new function which allowed for the shear modulus predictions to approach infinity
as the volume fraction approached its maximum, their modification was shown to
improve accuracy of E11 at high volume fractions [13]. An important flaw in the
Halpin-Tsai equations is its inability to accurately predict the transverse Poisson’s
ratio. It has been shown that predictions can be as much as three times higher than
6
2.1.2 Mori-Tanaka Model
In 1957, Eshelby developed a structural model for a single ellipsoidal inclusion within
an infinite matrix [15]. Mori and Tanaka extended Eshelby’s single inclusion to one
with multiple inclusions which could have stress and strain interactions [4]. This
interaction between inclusions has been ignored by Eshelby and therefore lead to
poor solutions at high volume fractions. The main assumption made by Mori and
Tanaka was that a fiber in a concentrated composite sees the average strain of the
matrix [8]. The Mori-Tanaka model gives a much better prediction for ν23 and slightly
better predictions for all other elastic properties when compared to the Halpin-Tsai
model [8].
2.2 Numerical Property Prediction
Several numerical methods such as the finite element method and boundary element
method have also been widely explored in the literature for predicting the material
properties of composites. These methods take on a more brute force technique of
accurately predicting the material properties of composites by discretizing the do-
main of interest. Numerical approaches allow for the representation of very complex
geometries while avoiding the simplifying assumptions required by micromechanical
models.
Despite these advantages, there are problems associated with employing numerical
methods when computing elastic properties. First, the domain must be discretized
7
through a meshing procedure. This introduces potential issues associated with mesh
stability and accuracy and also computer memory constraints. Mesh stability is very
important in the convergence of the model’s solution. Also, because of the geometries
inherent to fiber composites, tetrahedral elements must be used in order to mesh the
domain. These elements are constant strain elements requiring a very fine mesh
to achieve an accurate solution. This also emphasizes the constraint of computer
memory needed. Larger models require much more memory as well as much more
time to solve the finite element or boundary element equations.
2.2.1 Finite Element Method
The finite element method has been used in predicting material properties of com-
posite materials which can require significant computational resources, especially for
large problems. To reduce the required computational resources, representative vol-
ume element (RVE) techniques have been developed. By assuming that the composite
is periodic at some size level, the problem can be solved for a smaller RVE of the com-
posite which is often contain properties which are nearly equivalent to the properties
of the entire composite. Finite elements are used to fill RVEs in order to attain the
solutions. Introducing even more simplicity, RVEs have even been assumed to be as
small as one fiber or part of a fiber in order to decrease the problem size [1,8,16,17].
RVE techniques require similar underlying assumptions of regularity and uniformity
that the analytical micromechanical models include, however they can give insight
that was previously missing.
Because the size of the RVE is important for approximating the global elastic
8
properties, much research has been performed to investigate it [18–23]. As few as 30
fibers has been found to accurately reflect the global solution within a few percent.
Hine, et al. [21] concluded that 100 fibers gave accurate solutions with very little
standard deviation. These were very promising studies showing that a supercomputer
is not required to utilize this method.
Another area of interest investigated by Hine, et al. [21] was effects due to inherent
fiber length distributions. In manufactured composites, each fiber is slightly differ-
ent in length and diameter and the affect of these distributions on overall material
properties is of interest. In Hine, et al. [21], over 27,000 fibers of an injected moulded
composite part were measured in order to attain length and orientation distribution.
These were used to seed a Monte Carlo procedure with 100 fibers. Hine, et al. [21]
concluded that using the average length of the sample in the analysis could replace
seeding a length distribution for the analysis.
Similar to Hine, Gusev, et al. [14] considered the role of distributions of the fiber
diameter and spacial orientation with respect to unidirectional short fiber compos-
ites. An ultrasonic velocity measurement procedure was developed to obtain elastic
constants of the sample. Image analysis was used to measure fiber diameters. Once
again, the data seeded a Monte Carlo procedure. This study also investigated three
different packing schemes: random, hexagonal array, and square array. Like Hine,
et al. [21], the distribution had little effect on the results as long as the mean was
known. It was found that the packing scheme of the fibers did have significant effects
on the transverse material properties such as Young’s modulus, shear modulus, and
9
Poisson’s ratio. These results were compared to results computed with micromechan-
ical models where it was determined that the micromechanical models give unreliable
predictions for transverse material properties. This is because they don’t account for
the spatial distribution of the fibers within the model [14] .
In the late 1990’s, Gusev also developed a multiple inclusion RVE finite element
method for predicting properties of composites that are assumed to have some degree
of randomness to the arrangement of fibers [19, 20]. Gusev fills a unit cell randomly
with a Monte Carlo algorithm to distribute the inclusions throughout the RVE in a
manner similar to that found elsewhere [14, 21, 24–28]. Gusev considered spherical
inclusions and the process has even been used for packing aligned fibers in RVEs, but
becomes incredibly inefficient for misaligned fibers with any relatively high volume
fraction. Lusti, et. al. [24] modified Gusev’s random packing procedure by increasing
the size of the RVE until a certain number of fibers were included and not overlap-
ping. This resulted in a diluted volume fraction of around 0.1%. The size of the RVE
was then reduced by systematically moving the already placed fibers without chang-
ing their orientation. This allowed them to increase volume fractions up to around
8.0% which was previously impossible. The resulting domain was then meshed with
tetrahedral elements to capture the complex geometries of the model.
Several methods of simplifying the meshing procedure have been developed. In
order to reduce computational expenses, Witcomb, et al. [29, 30] have used a local
and global mesh combination to evaluate very fine microstructures. Local analyses
are performed within the refined mesh region and are based upon displacements and
forces from the global mesh. This translation of force between different meshes of
10
different element size becomes difficult since the sizes and number of nodes are so
different creating problems in attaining a reliable solution.
Zeng, et al. [31] also investigated the simplification of the meshing process. They
divided the RVEs into a regular array of subcells and used Gauss quadrature to
solve for the stiffness tensor. They defined materials by defining properties at each
Gauss point rather than for each element. If the Gauss point were to lie in the fiber,
then fiber properties were used and if it were not lying in the fiber, then matrix
properties were used. Out of plane modulus was found to be stiffer than that found
in literature [31]. Unfortunately, Zeng, et al. [31] ignored the resulting discontinuities
in the strain field within each element.
This research will incorporate a very similar finite element method with uniform
elemental arrays using Gauss quadrature to analyze the long fiber composites. Dis-
continuities in the strain field within each element is accounted for with a correction
factor. Long fibers, which have yet to be studied, will be investigated with the uniform
mesh finite element method.
2.2.2 Boundary Element Method
Rather than evaluating the entire three-dimensional solid domain of a model like
the finite element method, the boundary element method (BEM) only meshes the
surfaces of the model. This is done to solve the partial differential equations of the
continuum mechanics through surface integrals or boundary integrals. The bound-
ary element method has a significant computational advantage over finite elements
in that the number of degrees is much smaller. This corresponds to fewer equations
11
and unknowns in the solution, requiring less computational resources to solve the
problem [32]. In the mid 1990’s, Papathanisiou and Ingber [13,32,33] modeled short
fiber composites successfully. These models included as many as 200 fibers. With
the use of the boundary element method and a parallel supercomputer, they were
able to study composites which could not be studied by standard micromechanical
models or single fiber finite element models. The conclusion was that these simplified
micromechanics models suffice at low volume fractions, but lacks accuracy at higher
volume fractions. Also, they showed that aligned fibers show an increase in longitudi-
nal modulus compared to randomly distributed alignments [32]. One drawback of this
research was the exclusion of the idea of a RVE. This lead to boundary interactions
producing artificial fiber alignment in the model. Other assumptions were made with
respect to the matrix being incompressible and fibers being rigid in order to simplify
the boundary integral calculations.
More recently, Nashimura [34] introduced a new method for solving these bound-
ary integrals. The fast multipole method (FMM) decreases the solution time greatly
compared to the boundary element method. Solution time was reduced from O(N2)
for the BEM to O(N) for the FMM where N is the number of equations in the prob-
lem [35]. Lui, et al. [35] has evaluated RVE models with over 5800 fibers and 10
million degrees of freedom. Chen and Liu also developed a BEM with quadratic ele-
ments that can represent the matrix and fiber as elastic materials [36]. Their model
size was limited to a few thousand elements due to the use of the conventional BEM
solution with their quadratic elements [36].
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METHODS
This chapter explains in detail the procedures and methods used to perform the uni-
form mesh finite element method, including the finite element derivation and bound-
ary conditions applied. Halpin-Tsai and Tandon-Weng micromechanical models will
also be explained for comparison to the finite element method. This chapter also ex-
plores the method of extracting a model from micro computed-tomography (CT) data
scans. These image CT scans were received from Oak Ridge National Laboratory in
Tennessee. They were then modified by cropping, thresholding, and morphological
operations such as erosion and dilation in order to extract fiber geometry information
for uniform mesh procedure.
3.1 Finite Element Model
A uniform mesh finite element method is investigated and developed in order to
predict material properties for the fiber composites presented. A regular array of finite
elements are used, thus relieving the need for a robust meshing algorithm. Instead,
an increased number of Gauss points are employed in calculating the stiffness matrix
where the spacial distinction between fiber and matrix is decided and incorporated
into the Gauss quadrature. The material properties of the composite constituents are
assigned to each Gauss point depending on its spacial relation to the fibrous model.
The advantage of this procedure is that increasing accuracy requires only to increase
the number of Gauss points, not the number of elements, therefore not increasing
13
the degrees of freedom of the system and ultimately allowing for very complex fiber
models to be able to run on a desktop computer.
A special note of acknowledgement is needed for Elijah Caselman’s work on the
uniform mesh finite element method. Much of the derivations provided below are
credited to his thesis [37].
3.1.1 Finite Element Derivation
Hamilton’s principle of representing the differential equations of motion as an equiv-
alent integral equation can be shown to formulate the finite element equations [38].
Hamilton’s principle states
δLdt = 0 (3.1)
where δ is the variational calculus operator and L is the Lagrangian [38] which can
be written as
L = T − Πp = kinetic energy - potential energy (3.2)
The potential energy can be shown as a difference between strain energy Π and work
done Wp as
14
tTUdS (3.5)
where ε is the strain vector, σ is the stress vector, b is the body force vector, U is the
displacement vector, t is a vector of surface tractions, V is the volume of the body,
and S is the surface of the body.
A linear relationship between stress and strain may be defined by Hooke’s law as
σ = [C]ε (3.6)
In the case of an isotropic material, the stiffness tensor [C] is symmetric and defined
with two unknowns: Young’s modulus E and Poisson’s ratio ν as seen below [39]
[C] = 1
0 0 0 2(1 + ν) 0 0
0 0 0 0 2(1 + ν) 0
0 0 0 0 0 2(1 + ν)

−1
(3.7)
By combining Equations 3.4 and 3.6, applying the variational operator δ, and sum-
ming, the equation
is obtained. Applying the variational operator δ to Wp gives
δWp =
tTδUdS (3.9)
After combining Equations 3.1, 3.2, 3.3, 3.8, and 3.9, Hamilton’s principle can be
shown as
tTδUdSdt = 0 (3.10)
In order to integrate over the domain V in Equation 3.10, it is divided into Ne sub-
domains often called finite elements and the displacement vector U over the elemental
domain V (e) is interpolated as
U =

= [N ]d(e) (3.11)
where d(e) is the elemental displacement vector and [N ] is the matrix of shape func-
tions that relates the elemental displacement field to the nodal displacements. There
are 8 shape functions for the 8-noded brick element that will be used in the uniform
mesh finite element analysis which are written as
16
(3.12)
The shape functions are shown as functions of the natural element coordinates ξ,η,and
ζ. Each natural coordinate ranges from -1 to 1 and is zero at the center of the brick
element. For rectangular elements with edges parallel to the global coordinate axes,
the relation between global coordinates and natural coordinates is
x = x + ξ lx 2
y = y + η ly 2
(3.13)
z = z + ζ lz 2
where x,y, and z are the coordinates of the elements centroid and lx, ly, and lz are the
respective edge lengths of the element. It can then be shown that the infinitesimal
physical coordinates can be related to the infinitesimal natural coordinates as [39]
dx = ∂x
∂ξ dξ =
lx 2
dV (e) = dxdydz = lx 2
ly 2
lz 2
dξdηdζ (3.15)
The strain vector ε can be written in the form of a displacement gradient and then
separated into a matrix of partial differential operators and a displacement vector as
ε =
= [B]d(e) (3.16)
where the elemental strain-displacement matrix [B] for the element is written as [39]
[B] = [[B1][B2] . . . [B8]] (3.17)
where each [Bi] is defined as
[Bi] =
18
Combining Equations 3.10, 3.11, 3.16 and summing over all elements, we can now
express Hamilton’s principle as
Ne∑ ∫ t2
− ∫ (e)
S
δd(e)T
tT [N ]dS(e)dt = 0 (3.20)
The next step in solving the equation is defining the elemental stiffness matrix
and elemental force vector as
[K(e)] =
P(e) =
tT[N ]dS(e) (3.22)
respectively. Hamilton’s principle can now be written in terms of a forcing vector,
displacement vector, and stiffness matrix as
Ne∑∫ t2
) dt = 0 (3.23)
where the only non-trivial solution would be when [K(e)]d(e) − P(e) = 0. A simpler
representation of this is presented here
[K]D = P (3.24)
where [K] is the global stiffness matrix, D is the global displacement vector, and P
is the global force vector.
19
After combining 3.21, 3.15, [K(e)] can be evaluated using Gauss-Legendre quadra-
ture with Ngp Gauss points, ξi, ηi, ζi, and weights, Wi, through
[K(e)] =
ly 2
lz 2
dξdηdζ (3.25)
ly 2
3.1.2.1 One Dimensional
Conventional finite elements are typically limitted to only one material per element.
The elements used in this analysis must be modified to account for the possibility of
having 2 separate materials within one element. Multiple materials within a single el-
ement will be accomodated by applying a correction factor to the strain-displacement
matrix [B] [37].
The strain-displacement factor will first be explained through a one dimensional
example of two rods in series, one having matrix properties Em and Lm, and the other
with fiber properties Ef and Lf . In this analysis, E is the modulus and L is the rod
length. A is the cross sectional area which is the same for both fiber and matrix. It
can be shown that the force F in both springs is [37]
F = EmA
(u1 + u2) (3.28)
by requiring the total length to equal the sums of the individual lengths. This ensures
20
continuity in the model and it can be shown that
Eeff = EmEf
where β is the fiber length fraction defined as
β = Lf
Leff
(3.30)
(3.31)
The strain is equal to the strain-displacement matrix [B] times the elemental
displacement vector as shown in Equation 3.16. Therefore, in order to account for
various strain values within a single hybrid element, a correction factor is applied to
the strain-displacement matrix [37]:
[B] = α[B] (3.32)
where if the Gauss point lies within the matrix, the corrections factor is [37]
where α
α = Ef
Emβ + Ef (1− β) (3.33)
and if the Gauss point lies within the fiber, the correction factor is [37]
α = Em
21
3.1.2.2 Three Dimensional
The one-dimensional procedure may be extended to three dimensions by searching
all Gauss points ”in line” with the Gauss points and obtaining a fiber length fraction
for each direction as
(3.37)
where W ′ is the Gauss weights at each point lying in a fiber. Correction factors are
obtained for each direction and for each Gauss point in each element and the factors
enter the strain-displacement matrix calculation in Equation 3.18 as [37]
[Bi] =
3.1.3 Boundary Conditions
It has been shown that effective properties attained from a model depend on the
boundary conditions imposed [40]. Namat-Nasser [40] found that uniform traction
boundary conditions result in a lower bound on properties while uniform displacement
boundary conditions result in an upper bound on properties. This is reflective of the
bound resulting from the Voigt constant strain assumption and Reuss constant stress
assumption in previous models.
Appropriate periodic boundary conditions must ensure each RVE has the same
deformation mode such that there is no separation or overlap between adjacent RVEs
[16]. Essentially, RVEs must be periodically stackable at all times in each of the
coordinate directions. These conditions are met by this displacement field
ui = εikxk + u∗i (3.39)
where ui is the ith component of the displacement vector, εik is the average strain
tensor, xk is the kth component of the coordinate vector, and u∗i is the ith component
of the periodic displacement vector. It follows that periodic displacement boundary
conditions for RVEs are expressed as [16]
uj+ i = εikx
uj− i = εikx
j− k + u∗i (3.41)
Where boundary surfaces are referred to as the j+ and j - to indicate the positive
and negative xj surfaces. The difference between periodic surface nodes must remain
23
constant so that subtracting these displacements from positive and negative surfaces
must always yield a constant as seen below.
uj+ i − uj−
k ) = cj i (3.42)
Note that cj i = 0 yields no relative displacement between the sides of the RVE.
Alternatively, axial extensiona dn shear are imposed through non-zero cj i .
3.2 Composite Micromechanics Models
3.2.1 Halpin-Tsai Model
By combining the self-consistent approach of Hill with the solutions of Hermans and
making a few additional assumptions, Halpin and Tsai provide a simpler analytical
form for predicting material properties of fiber composites [5, 10] . The Halpin-Tsai
equations need only one equation to find all the composite moduli and the longitudinal
Poisson’s ratio is simply found from the rule of mixtures. It should be noted that
the Halpin-Tsai equations are partially empirical where one parameter ζ is found by
fitting the equations to numerical results. The Halpin-Tsai equations for the elastic
constants are given by [5]
P
Pm
ν23 = E22
24
where P represents any moduli and Pm and Pf are the corresponding moduli of the
matrix and fiber. The parameter ζ is dependent upon the moduli being calculated as
shown in Table 3.1 where the value of ζG12 proposed by Hewitt and Malherbe [11] is
shown.
P Pf Pm ζ E11 Ef Em 2(l/d) E22 Ef Em 2 G12 Gf Gm 1 + 40v10
f
Table 3.1: Halpin-Tsai parameters used for short fiber calculations [1]
The bulk modulus of the matrix Km is found from the simple relationship between
material constants as Km = Em
3(1−2νm) .
3.2.2 Tandon-Weng Model
Tandon and Weng derive explicit expressions for the elastic constants of a short-fiber
composite using the Mori-Tanaka [4] approach. Their formulae for the plane-strain
bulk modulus k23 and the major Poisson ratio ν12 are coupled, and must be solved
iteratively [6, 8]. The Tandon and Weng calculations are given as
25
E22 = Em
G12 = Gm(1 + vf )
(3.49)
(3.50)
ν23 = E22
− 1 (3.52)
The equation for ν12 shown here was derived by Tucker and Liang [8] and provides a
non-iterative formula to the iterative equation of ν12 presented by Tandon and Weng.
The constants are found with Equations 3.53
A1 = D1(B4 + B5)− 2B2
A3 = B1 −D1B3
A5 = (1−D1)/(B4 −B5)
A = 2B2B3 −B1(B4 + B5)
The constants Bi and Dj are found from the following
26
B2 = vf + D3 + (1− vf )(D1S1122 + S2222 + S2233) (3.55)
B3 = vf + D3 + (1− vf )(S1111 + (1 + D1)S2211) (3.56)
B4 = vfD1 + D2 + (1− vf )(S1122 + D1S2222 + S2233) (3.57)
B5 = vf + D3 + (1− vf )(S1122 + S2222 + D1S2233) (3.58)
D1 = 1 + 2(µf − µm)/(λf − λm) (3.59)
D2 = (λm + 2µm)/(λf − λm) (3.60)
D3 = λm/(λf − λm) (3.61)
where µm, µf , λm, and λf are Lame’s constants for the matrix and fiber materials.
Lame’s constants are related to the Young’s modulus E and Poisson’s ratio ν by
λ = Eν
2(1 + ν) (3.63)
The Sijkl in Equations 3.54 are the Eshelby tensor components for a spheriodal in-
clusion defined as
}
where α is the aspect ratio of the fiber, and g is a parameter defined as
g = α
(α2 − 1)3/2
3.3 Digital Image Processing
Image processing can be defined as changing the nature of an image in order to
either [2]:
2.) Render it more suitable for autonomous machine perception
This thesis explores digital image processing which analyzes and modifies digital
images on a computer. A digital image differs from a photo in that x, y, and f(x, y) are
all discrete integers. Usually, these points in the image are called picture elements, or
more simply pixels having values that range from 1 to 256 with a brightness value from
0(black) to 255(white). The pixels surrounding any single pixel of interest constitute
28
its neighborhood. A neighborhood can be defined as a matrix of elements. For
example, a neighborhood could be a 3x3, 5x5, or 3x7 array of pixels. Neighborhoods
usually have an odd number of rows and columns so that the pixel of interest is
centered. If the neighborhood had an even number of rows or columns, it would be
necessary to specify which element would be the current pixel of interest [2, 41].
Understanding the basics of image data structures is important. First, a digital
image can be viewed as a matrix of values which correspond to some value of color
or intensity. The four basic types of images are binary, greyscale, true color (RGB),
and indexed. Binary images are black and white images in which the matrix of values
are filled with either an on or off setting. Greyscale images have a scale of shades
between 0(black) and 255(white). There are 256 possibilities because the data is
stored in 8 bits (8 bits with 2 possibilities per bit = 28 = 256 possibilities per pixel).
True color images are an extension of greyscale, but with shades of color divided into
3 layers of color: one layer of red, one layer of blue, and one layer of green. These
combinations give 16,777,216 color possibilities per pixel(2563) [41]. Lastly, indexed
images take advantage of the fact that most images do not have 16+ million colors.
For simplification and in order to save memory, an index of colors in the image is
used and the images are saved with this smaller index number as its data. This index
then refers to a certain color allowing for images to be saved using less space than
full 3 matrix layers of RGB color size.
In order to conveniently study digital image processing, the different algorithms
can be classified based on the nature of the task at hand. These categories are [2]
29
• Image Enhancement
• Image Restoration
• Image Segmentation
Image enhancement refers to modifying an image so that the result is more suit-
able for a particular application [2]. Examples include sharpening an out-of-focus
image, highlighting edges, improving contrast or brightness, and removing noise. Im-
age restoration refers to reversing damage done to an image by some known cause
such as removing a blur caused by a filter, removing optical distortions, or remov-
ing periodic interference. Finally, image segmentation involves subdividing an image
into constituent parts, or isolating certain aspects of an image. Examples of image
segmentation are finding lines or particular shapes in an image and identifying cars,
trees, buildings, or roads in aerial photography. It is important to note that all of
these image processing operations share common elements. The same algorithms may
be used for more than one category with the goal of the procedure determining its
purpose.
The rest of this section explains basic image manipulations that can effectively be
used to extract fiber/matrix data from the set of slices. Spacial filtering, cropping,
rotation, thresholding, and a few morphological operators like erosion and dilation
are explained in the following in more detail.
30
Spacial filters are usually associated with smoothing or edge detection algorithms.
They look at a neighborhood mask of pixels and modify the pixel of interest based on
a function of these neighborhood pixels. The mask is usually an array with an odd
number of rows and columns. As stated before, this allows for the neighborhood to
center on the pixel of interest. The combination of neighborhood mask and function
defines the filter. Figure 3.1 shows the basic procedure of filtering an image.
Figure 3.1: Performing a spacial filter [2].
One widely used filter is known as median filtering where the pixel of interest is an
average of its surrounding neighbors. Figure 3.2 shows the function of a pixel for a
3x3 masked median filter.
Figure 3.2: Algorithm for a 3x3 median filter [2].
In this figure, e is the pixel of interest that is changed from the function and mask.
Another important category of filters is known as frequency filters. These fre-
quency filters analyze how quickly and how often the pixels change intensity within
an image and modify pixels based on the information [2]. High pass filters leave high
frequency components and only affects low frequency parts of an image. Conversely,
low pass filters leave low frequency components and only affect high frequency por-
tions of the image. High pass filters provide excellent edge detection within images.
A few high pass filters used for this include the Laplacian and Laplacian of Gaussian
filters. The Laplacian filter is a 2-D isotropic measure of the 2nd spatial derivative of
an image. The Laplacian in 2-D is defined as
L(x, y) = ∂2I
∂y2 (3.66)
where I(x, y) are the intensity values of the image. Since an image is discrete, the
mask used is an approximation of this function. The Laplacian of Gaussian is the
same filter applied after a Gaussian filter has been performed. This is done to reduce
the sensitivity inherent to the Laplacian filter. The Gaussian filter is an averaging
technique based on a nonuniformly weighted mask that approximates the Gaussian
32
distribution function discretely over the mask.
There are many other filters with various advantages and disadvantages. The
discussion here is limited to the Fourier Transform and Discrete Fourier Transform
since they are widely used in image processing image processing. These transforms
allow for tasks which would be impossible to otherwise perform as their efficiencies
allow other tasks to be performed more quickly. The Fourier Transform provides,
among other things, a powerful alternative to linear spatial filtering. It is also more
efficient to use the Fourier transform than a spatial filter for a large filter. The Fourier
Transform allows the isolation and processing of particular image frequencies, and can
therefore perform high and low filter passes with great precision and speed.
There are two items to keep note when performing any type of filtering: edge
decisions and using a shadow image to apply a filter. Because masks are not fully
defined near edges, deciding what to do with them becomes important. Sometimes
the edges are ignored all together performing the filter only on pixels who’s masks
are fully defined. Another popular way of avoiding these edge effects is assuming the
undefined portions of the mask to be zero. Second, the shadow copy is a simple but
crucial step that must be mentioned. In performing filtering, a new image must be
created based on the old one being filtered. The old image must not be updated as
in some other processes or the filtered data would affect the unfiltered data’s filtering
process.
33
3.3.2 Cropping and Rotation
These simple image editing techniques may seem trivial but are very important in
reducing memory requirements and computational time as well as establishing a co-
ordinate system that is consistent with the scanned data. Cropping removes rows or
columns of the image within a specified region and rotation moves pixels so that the
region of interest is rotated. An example of a simple crop can be seen below.
Figure 3.3: Simple cropping of an image.
3.3.3 Thresholding
Thresholding is a process by which you limit values of an image [3]. For example, a
256 or 8 bit (28 = 256) per pixel gray-scale image could have a thresholding procedure
applied where all the dark regions or possibly all of the white regions are removed.
The procedure for choosing the value at which to threshold is where this procedure
can become very complicated. There are many statistical and weighted methods that
can be used for determining a threshold value. The procedure is as simple as an if
statement in programming language. An example of thresholding a greyscale image
34
is shown in Figure 3.4.
Figure 3.4: Simple threshold of an image of rice (Source: MATLAB, R2007b, The MathWorks, Natick, MA).
3.3.4 Morphological Operations
Morphological analysis started in the late 1960’s. It uses Minkowski algebra to ma-
nipulate classical two valued black and white images [41]. These set-based operations
are used to enhance photos to extract data that may otherwise be overlooked. Struc-
tural elements are defined and used as sets for the operations. They can be any shape
with round or square being most popular, and are similar to masks described above,
but are not normally rectangular in shape. In the case of this research, the focus will
be on cleaning up CT data TIFF images in order to achieve a binary representation
of each slice that depicts fibers and matrix. Figure 3.5 shows an image and simple
morphological operation used to modify its appearance. By using multiple successive
iterations of morphological operations, it is possible to remove noise and retain the
fiber geometry.
It can be seen in Figure 3.5 that the procedure and order of operations, give much
35
Figure 3.5: Example diagram of morphological operations [3].
different results. Figure 3.6 is another example of the implications of structural
element size.
Clearly if the structural element chosen is large, too much data is removed and made
impossible to later retrieve. This is why relatively smaller elements are used with
multiple iterations in order to achieve larger dilations or erosions. Figure 3.7 is another
example of how iterations of erosion and dilation can remove noise but keep inherent
pieces of the image. The pepper like noise in Figure 3.7 can be removed without
eliminating the rest of the image.
3.3.4.1 Erosion
Erosion removes pixels from an image. For binary images, erosion is performed by
turning a pixel from ON to OFF by some criteria based on the values of neighboring
36
Figure 3.6: Panda image example illustrating importance of structural element size [3]
Figure 3.7: Example of noise removal by erosion and dilation [3].
pixels. Erosion is defined by the intersection of an image and a structural element as
AªB =
A−b (3.67)
where ª is Minkowski subtraction, A is the image and B is the structural element
of erosion, and b is the movement vector for A [41]. In a sense, if B does not ”fit”
somewhere inside A, then that pixel of the image is turned OFF. This process is
sometimes referred to as image shrinking [3].
37
3.3.4.2 Dilation
Dilation adds pixels to an image. This is done by turning a pixel from OFF to ON
by some criteria based on the values of neighboring pixels. Dilation is defined by the
union of an image and a structural element as
A⊕B =
b∈B
Ab (3.68)
where ⊕ is Minkowski addition, A is the image and B is the structural element of
erosion, and b is the movement vector for A [41]. This process is sometimes referred
to as image growing [3]. Dilation and erosion are essentially ”inverse” operations of
one another. Their relationship can be shown as
AªB = A⊕ B (3.69)
A⊕B = Aª B (3.70)
where the complement of an erosion is equal to the dilation of the complement [2].
A simple example of erosion between A and B is shown below in Figure 3.8. The
pixels are either added by dilation or removed by erosion.
38
Figure 3.8: Example of erosion and dilation shown at the pixel level. (Source: http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip-Contents.html)
39
SIMPLE CURVED LONG FIBER MODELS
In this chapter, the uniform mesh method is applied to predict the elastic properties of
simple long fiber models. The models are all regularly packed arrays of long fibers that
are 1mm long with different curvatures. The fibers are laid in a planer orientation in
order to simplify the procedure and to provide for a better comparison among models.
The results of the uniform mesh method is compared with results in the literature
where possible. In addition, FEA results will be compared with the Halpin-Tsai and
Tandon-Weng micromechanical models that are used for short fiber composites.
Each model is defined as a planar array of regularly stacked and positioned fibers.
The models are different only in fiber curvature where a simple quadratic function
is used to define fibers having a constant curvature. An engineering definition of
curvature was used here as the second derivative of the function that defines the
fiber’s geometry. The models used here appear in Figures 4.1-4.4 and have a fiber
volume fraction of 0.1535 with the same RVE, fiber length, and number of fibers.
The x direction is the axial direction of the fibers, y direction is normal to this in the
plane of fibers, and z direction is normal to the fiber axis and plane.
40
Figure 4.1: Model of Long Fiber with curvature of 0.
4.1 Effective Elastic Properties
The elastic properties are predicted by the uniform mesh finite element method,
Halpin-Tsai equations, and a modified Tandon-Weng equation. The process to predict
material properties for the curved fibers with the uniform mesh finite element method
is discussed in detail below.
ABAQUS (Simulia, Providence, RI) solves a system based on an input file that
contains the nodal coordinates, connectivity matrix for elements, materials, boundary
41
conditions, and requested output. Fortran was used to write the input file for the
regular array of parallelpiped elements used in the analysis. The mesh used for the
regular array of simple curved long fiber is shown below
where there are 120 elements in the x-direction, 40 elements in the y-direction, and 10
elements in the z-direction. These models were analyzed with 48,000 elements using
6 Gauss Points for integration in each direction. This problem had 163,722 degrees
of freedom and was solved with ABAQUS 6.6-3.
Because the uniform mesh did not define the fibers, an algorithm was needed to
42
Figure 4.5: Mesh used for all simple curved long fiber models.
distinguish between fiber and matrix within the elements. The Gauss point properties
were determined by a simple algorithm based on the definitions of the fibers
y = f(x) = c
2 (x− xf )
2 + yf ; x ∈ (xmin, xmax) (4.1)
where c is the curvature of the fibers, xf is the x location of the middle of the fiber,
and yf is the y location of the center of the bottom most portion of the fiber. xmin and
max are predetermined based on the fiber length of 1mm. The equation for solving
for these two value is given as
s =
∫ xmax
xmin
√ 1 + [f ′(x)]2dx (4.2)
For the straight fiber case, xf and yf its centroid. The Z direction is not needed since
the fiber lies on the xy-plane, therefore the z-location is known.
43
Since the length of the fiber is known, its ”centroid” is defined, and the function
for the fiber is specified, testing a simple geometric calculation based on an ellipsoidal
cross section of a round fiber cut at an angle could give enough information to deter-
mine if the Gauss point is in the fiber. The model is sliced parallel to the yz-plane to
determine a fiber cross section ellipsoid can be determined create a condition for the
fiber to lie inside or out of the fiber. The conditions for being in a fiber are
(zgp − zf ) 2
(ygp − yf ) 2
b2 ≤ 1 (4.3)
x ≥ xmin (4.4)
x ≤ xmax (4.5)
where zgp and ygp are the Gauss point locations and a and b are the minor and major
axes of the cross sectional ellipse, respectively.
After the algorithm for defining fibers inside the uniform mesh was developed, an
ABAQUS UEL user subroutine developed previously by Caselman [37] was used to
implement the unique element type described in Chapter 3. The routine is called
at the beginning and end of each solution iteration. Input to the subroutine is the
current nodal displacement vector, nodal coordinates, and material properties and
the subroutine returns the elemental stiffness matrix and forcing vector. The periodic
boundary conditions are definted through the ABAQUS ”*EQUATION” command.
ABAQUS then assembles the global matrices and vectors and solves the system of
equations for the unknown nodal displacements.
In order for the UEL subroutine to return the correct elemental matrices and
44
vectors, it must include the functions definted above in Equations 4.3 - 4.5. The
correction factors in Equations 3.33 and 3.34 are based on these Gauss point material
properties and are used to calculate the elemental stiffness matrices in Equation 3.25.
The elemental force vector is found simply by multiplying the calculated stiffness
matrix with the given elemental displacement vector from ABAQUS.
Since the uniform mesh method uses a modified linear 8-node brick element, stan-
dard parameters like stress and are not easily output through the solution. In order
to easily gain these plots, standard ABAQUS C3D8 8-node, 3 degrees of freedom per
node, brick elements are overlayed with the same nodal coordinates and connectivity
matrix as in Caselman [37]. So that these would have a negligible affect on the results,
they were given modulus values on the order of 108 times smaller than the fiber or
matrix. Unfortunately, the strains or stresses are then only calculated at the C3D8
Gauss points so all contour plots to follow do not show as much detail as is in the
analysis.
Once the finite element solution is performed, the resulting reaction forces are
used to compute the effective material properties for the RVE. The average strain is
defined as
εij = 1
εijdV (4.6)
where V is the volume of the RVE. The strain tensor εij can be written in terms of
displacement as
2 (ui,j + uj,i) (4.7)
The average strain in Equation 4.6 is transformed to a the boundary integral using
Equation 4.7 and the Gauss Theorem as [1]
εij = 1
(uinj + ujni)dS (4.8)
where nj is the jth component of the unit normal vector to the boundary surface S
of the RVE. For a parallelepiped element in which the boundary faces are normal to
the coordinate axes, the normal will have only one non-zero component and Equation
4.8 reduces to [16]
lilj (4.9)
where li refers to the length of the RVE in the xi direction, and cj i is the constant
defining the periodic boundary condition given in Equation 3.42. Therefore, the
average strain is found from the size of the RVE and the periodic boundary conditions
above.
In a similar manner, average stress is defined in terms of the stress tensor σij as
46
σij,j = 0 (4.11)
given that there are no body forces. Using the equilibrium equation it can be shown
that [42]
= σij (4.12)
Combining Equations 4.10, 4.12 and again using Gauss’s theorem, it can be shown
that
σikxjnk dS (4.13)
By the definition of periodic boundaries, the stress at two corresponding points
on opposite surfaces must be equal. Similar to the derivation of Equation 4.9, the
average stress can be shown as [16]
σij = 1
]
47
when m 6= j, x+ j = x−j , and for m = j, then x+
j − x−j = lj and therefore [16]
σij = lj V
Sj
(4.15)
where Rij is the sum of the reactions forces on the boundary face Sj where j does
not indicate summation.
Once the average stress and strain have been obtained through Equations 4.9and
4.18, the effective elastic properties are evaluated by
σ = [C]ε (4.16)
where [C] is the average or effective stiffness matrix of the composite, σ is the averaged
stress vector obtained from 4.18, and ε is the averaged strain vector from 4.14. The av-
eraged stress vector is related to the stress tensor by σ = [σ11, σ22, σ33, σ23, σ31, σ12] T .
The average strain vector and tensor are related similarly. For an orthotropic compos-
ite, the average stiffness tensor is related to the average material properties through
the following

−1
(4.17)
From Equation 4.17 it is shown that for an orthotropic material there are nine inde-
pendent material properties (E11, E22, E33, ν12, ν23, ν13, G12, G23, G13). Therefore, nine
independent equations are needed to solve for the nine independent material proper-
ties. The nine equations are obtained from six independent strain conditions defined
48
i = 0 (4.18)
i = 0 (4.19)
i = 0 (4.20)
2 = 0.025lx, all other cj i = 0 (4.21)
set 5 : c3 2 = 0.025lz, c2
3 = 0.025ly, all other cj i = 0 (4.22)
set 6 : c3 1 = 0.025lz, c1
3 = 0.025lx, all other cj i = 0 (4.23)
The six strain sets represent three uniaxial extension conditions and three pure
shear conditions. From the three uniaxial extension conditions nine nontrivial equa-
tions will be obtained, only six of which are independent, and from the three pure
shear conditions the three remaining independent equations will be obtained.
To ensure the effective stiffness matrix is of the form in equation 3.7, Lagrange
multipliers are used as in Caselman [37] when solving equation 3.6 to impose the
necessary symmetry constraints. This is done by first forming average stress and
strain matrices from the average stress and strain vectors obtained from the six finite
element runs so that Equation 3.6 becomes
[σ] = [C][ε] (4.24)
By multiplying both sides of Equation 4.24 by the transpose of the average strain
matrix [ε]T the following unconstrained least squares equation is obtained
[ε]T [σ] = [C][ε]T [ε] (4.25)
49

b = [A]C (4.27)
where i and j are the components of the stress and strain matrices and i, j ∈ 1, 2, .., 6.
The 6 × 6 orthotropic stiffness matrix contains 12 non-zero constants. Due to
symmetry, the number of constants reduces to 9 as described above. Therefore, 27
constraints must be applied to the solution process. These constraint equations are
combined with Equation 4.27 as follows (see e.g. [43])
b
0
(4.28)
(4.29)
where λ is a vector of Lagrange multipliers, and [X] is a matrix of symmetry and
zero constraints. Therefore, the modified C vector is computed by solving
C
λ
4.2 Computed Results
The ability of the uniform mesh finite element model to predict properties for long
fibers will now be investigated. The results for the long fiber models are compared
to the Halpin-Tsai and modified Tandon-Weng models. The material properties used
will be an idealized set following Tucker and Liang [8] where Ef/Em = 30 as given in
Table 4.1.
Table 4.1: Elastic properties of matrix and fiber
The results for the Halpin-Tsai and Tandon-Weng models for straight fibers as
well as the results for the set of long fiber models is shown below in Table 4.2 and ??.
Elastic Halpin− Tandon− Curvature constants Tsai Weng of 0
E11/Em 4.68 5.18 4.76 E22/Em 1.48 1.43 1.68
ν12 0.35 0.53 0.38 ν23 0.60 0.53 0.36
G12/Em 0.49 0.49 0.57 G23/Em 0.47 0.47 0.52
Table 4.2: Effective property results Halpin-Tsai and Tandon Weng models with comparison to straight fiber (curvature=0) finite element model.
Table 4.2 shows good agreement for E11 in comparison to the Halpin-Tsai results,
but the Tandon-Weng result are 9% higher. This could be due to the fact that the
aspect ratio is much higher than allowed for these micromechanical models. The
51
shear terms of the uniform mesh method are within 15% of both micromechanical
models. The Poisson’s ratio ν23 underpredicts the Halpin-Tsai results by nearly 45%,
however, similar trends related to the Halpin-Tsai equations appear elsewhere [8,21].
The Tandon-Weng model is also higher for ν23 as compared to the uniform mesh.
Note, however, that ν12 is in good agreement between the uniform mesh and Halpin-
Tsai model, but the Tandon-Weng results are nearly 40% higher than either of these.
Next, related calculations based on the short fiber suspension mechanics models
of Advani and Tucker [7] and also Jack and Smith [44] are applied to the curved fiber
models presented. These short fiber models depend on unidirectional fiber compos-
ite properties which can be evaluated with any micromechaniics model such as the
Halpin-Tsai or Tandon-Weng models. The orientations tensors aij and aijkl (see Ad-
vani and Tucker [7]) are employed to account for the curvature of the fibers through
a length averaged orientation tensor evaluated through
aij =
pi(s)pj(s)pk(s)pl(s)ds (4.32)
where Nf is the number of fibers, LI is the fiber length, and the pi are components
of the unit vector which define the fiber direction [7,44]. Results for the Halpin-Tsai
and Tandon-Weng models for the curved fiber models are compared with the uniform
mesh finite element analysis in Tables 4.3, 4.4, and 4.5.
52
Elastic Halpin− Tandon− FEA constants Tsai Weng
E11/Em 4.07 4.43 3.43 E22/Em 1.44 1.40 1.60 E33/Em 1.52 1.46 2.18
ν12 0.48 0.65 0.44 ν21 0.17 0.20 0.20 ν13 0.28 0.43 0.26 ν31 0.10 0.14 0.16 ν23 0.56 0.50 0.33 ν32 0.59 0.52 0.45
G12/Em 0.70 0.71 0.53 G23/Em 0.47 0.47 0.50 G31/Em 0.48 0.48 0.53
Table 4.3: Short fiber-based results using Halpin-Tsai and Tandon Weng models with comparison to finite element model with a curvature of 1.
Similar trends shown in the results of Table 4.2 are also seen for all curved fiber
models. The Halpin-Tsai and Tandon-Weng models both predict higher moduli in all
directions for curvatures of 1, 2, and 4 except for the E22 predicted for a curvature
of 4. Again, this could be due to the fact that the aspect ratio is so high for the long
fibers. The shear moduli G23 and G31 of the uniform mesh method are within 15% of
both micromechanical models, but G12 is 30− 50% lower than the micromechanical
models. All Poisson’s ratios are computed with high errors indicating that traditional
micromechanical models are not applicable to the long fiber composites studied here.
53
E11/Em 3.16 3.40 2.56 E22/Em 1.46 1.46 1.56 E33/Em 1.61 1.51 2.16
ν12 0.58 0.70 0.47 ν21 0.27 0.30 0.29 ν13 0.22 0.34 0.24 ν31 0.11 0.15 0.20 ν23 0.48 0.45 0.30 ν32 0.53 0.47 0.42
G12/Em 0.95 0.99 0.62 G23/Em 0.47 0.47 0.53 G31/Em 0.48 0.48 0.58
E11/Em 2.31 2.47 1.97 E22/Em 1.75 1.84 1.62 E33/Em 1.73 1.58 2.15
ν12 0.53 0.60 0.45 ν21 0.40 0.44 0.37 ν13 0.28 0.35 0.24 ν31 0.21 0.22 0.26 ν23 0.38 0.40 0.27 ν32 0.38 0.34 0.36
G12/Em 1.12 1.17 0.66 G23/Em 0.47 0.47 0.55 G31/Em 0.48 0.48 0.55
54
From a comparison of the curved fiber analyses, it is seen that E11, ν13, and ν23
all decrease as the fibers curvature increases as shown in Figures 4.6 and 4.8. Even
a curvature of 1 decreases E11 by over 27.9% compared to straight fibers. Properties
which increase as curvature increases includes ν12, ν21, ν31, G12, and G23. The largest
increases between a curvature of 0 and 4 was ν21 and ν31 with a 184% and 100%
increase respectively.
4 .763
3 .440
2 .564
1 .971
0
1
2
3
4
5
6
E ff
e c
ti v
e P
ro p
e rt
E 11/E m
E 22/E m
E 33/E m
Figure 4.6: Elastic modulus property predictions for all curved fiber models.
Contour plots of Von Mises stress fields for each of the 6 loading cases for the
curvature of 1 model appears in Figures 4.9, 4.10, 4.11, 4.12, 4.13, and 4.14. These
are indicative of the other curved fiber model results and reveal the fiber locations
within the uniform mesh as well as the general stress state of the RVE.
55
0 .626 0 .607
G 12/E m
G 23/E m
G 31/E m
Figure 4.7: Shear modulus property predictions for all curved fiber models.
0 .134
0 .204
0 .285
0 .372
0 .135
0 .163
0 .199
0 .258
0 .000
0 .050
0 .100
0 .150
0 .200
0 .250
0 .300
0 .350
0 .400
nu21
nu31
Figure 4.8: Poisson’s ratio predictions for all curved fiber models.
56
Figure 4.9: Von Mises stress plot of simple fiber model with curvature of 1 during εxx.
Figure 4.10: Von Mises stress plot of simple fiber model with curvature of 1 during εyy.
Figure 4.11: Von Mises stress plot of simple fiber model with curvature of 1 during εzz.
57
Figure 4.12: Von Mises stress plot of simple fiber model with curvature of 1 during εxy.
Figure 4.13: Von Mises stress plot of simple fiber model with curvature of 1 during εyz.
Figure 4.14: Stress plot of simple fiber model with curvature of 1 during εzx.
58
Stress contour plots on the deformed mesh verify the periodicity in the solution
as seen in Figure 4.15. Arched deformations are seen in all curved models. The
straightening of the fibers during axial loading causes the model change shape and
begins arching the RVE. Figure 4.17 presents the model’s periodicity well.
Figure 4.15: Stress plot with deformation of simple fiber model with curvature of 1 during εxx.
Figure 4.16: Stress plot with deformation of simple fiber model with curvature of 1 during εyy.
59
Figure 4.17: Stress plot with deformation of simple fiber model with curvature of 1 during εxy.
60
COMPOSITE SAMPLE
For this research, Oak Ridge National Laboratory (ORNL) has provided the CT image
data in TIFF format of a fiber composite specimen. These CT images will provide a
means of distinguishing between fiber and matrix in the uniform mesh finite element
method in order to predict material properties for the ORNL composite. CT imaging
is used to obtain the geometric definition of fibers within a moded composite part.
Extracting the relevant data out of the scanned images required additional evaluation
in this research as the CT images can be grainy, distorted, and sometimes incomplete.
The fibers are relatively small compared to the resolution of what the scanners were
developed for - medical examinations of the human anatomy. Since fibers can be as
small as 10-20 micrometers in diameter, specialized micro-CT scanners must be used.
Even at these very high resolutions, the structural data is imperfect and needs to be
digitally enhanced and then extracted. A simple representation of the scanned stack
of data is shown below in Figure 5.1 where each slice represents a CT image.
Each layer of the stack is referred to as a slice and is regularly spaced apart from
the other layers. Each layer is a matrix of data scanned as a TIFF, DICOM, or BMP.
This information can be manipulated in order to retrieve the data stored within the
images. Some research is performed where the lengths or diameters are measured
from the scans, and these data points are used to seed Monte Carlo algorithms [21].
Because of the shear size of the model, this uniform mesh finite element model was
61
X
Y
Z
Figure 5.1: Simple model illustrating CT data stacks.
evaluated with a poor mesh size for the entire composite, and also split up by 3
in each direction giving 27 smaller pieces of the whole. This chapter analyzes and
discusses both models with the 3x3x3 split model first. The figures below give a
simple representation of the models. The first drawing, Figure 5.2 illlustrates the
size and relation of the two different approaches and the second, Figure 5.3 shows
the fibers segmented and modeled in 3D by using Amira (Visage Imaging, Carlsbad,
CA).
62
X
Y
Z
y = 816 px.
ORIGIN
5.1.1 Analysis Procedure
This section explains the image manipulations used to extract fiber/matrix data from
the set of CT slices obtained from ORNL. The CT data will then be used to determine
if the finite element Gauss points are within a fiber or not in order to implement
the UEL subroutine with ABAQUS as described in Chapter 4. Cropping, rotation,
thresholding, and the erosion and dilation morphological operators were used. The
first slice in the set of CT data from ORNL appears in Figure 5.4 in its original form.
The first step in the image processing for the TIFF CT data files was to crop the
images. This was done in order to save memory since each of the 696 slices analyzed
63
3
1
2
Figure 5.3: 3D Model of a region the size of one of the smaller models.
was over 11 MB in size. Also, the fact that most of the boarder data outside the
actual sample as seen in Figure 5.4 was not needed to define the finite element model.
The cropping operation was performed in MATLAB which provided a convenient
computational tool for manipulating pixel values.
Following the cropping operation, each grayscale TIFF image was thresheld in
order to further distinguish between the fibers and the matrix. Fibers were found to
have an intensity value of around 252, so a thresholding procedure was done with the
cutoff being 245. This produced a clearer binary image with some noise still present.
To further improve the quality of each image, morphological filtering processes were
employed which include both erosion and dilation. The morphological processes were
64
Figure 5.4: Slice 1 of 1200 provided by ORNL.
found to be very sensitive to the size of the structural element used since the image
resolution resulted in approximately 7 pixels across a fiber diameter. This relationship
of fiber diameter to image resolution limited the useful size of the smoothing mask.
An octagonal shape, diamond shape, square shape, and circular shape mask were all
examined, but only the smallest diamond shape consisting of 4 pixels gave reasonable
results. All other shapes were so large that the entire structures of the fibers were
lost in one iteration of erosion. Even though a very smooth definition of fibers could
be extracted, a few iterations of erosion followed by iterations of dilation resulted in
a very reasonable looking binary image of fiber and mesh compared to the original
image. The process of cropping, thresholding, eroding, dilating, and finally rotating
65
were all automated through the stack of data with MATLAB. Below, Figure 5.5 shows
the progression of these steps.
Figure 5.5: Showing the entire image processing procedure. a) after cropping b) after thresholding c) after morphological smoothing d) after final rotation and cropping
Figure 5.6 shows the iterations of the morphological steps in smoothing the image of
the first slice of data. It is clear that the noise left behind from the thresholding is
significantly reduced giving clearer representation of fiber and matrix geometry.
66
a b
c d
e f
Figure 5.6: An example showing the progression of morphological operations to remove noise from thresheld image 0001. a) original image, b) 1st erosion c) 2nd erosion d) 1st dilation, e) 2nd dilation, f)last dilation and final data image used for analysis.
67
Once all images had been processed, the binary pixel data of the entire model
composed of 696 slices which were each 852x816 pixels was arranged into a binary
column vector of data. The vector was arranged in a very systematic way as shown in
Figure 5.7 so that the Gauss point location algorithm could easily access component
data. This type of arrangement for the data allowed for x,y, and z coordinates to
relate to an index number for the vector of data.
Start
Finish
The index for pixel location is determined from
INDEX = (K − 1)× 695232 + (I − 1)× 852 + J (5.1)
where
68
lp = 230
98 10−6
where lp is the size of one side of the square pixels, and xo,yo, and zo are vectors from
the global coordinate origin to the coordinate origin of each smaller 3x3x3 geometry.
The FLOOR function is used in FORTRAN (The Fortran Co., Tucson, AZ) to round
down any real number to its closest integer value. Figure 5.8 shows the finite element
mesh used for each of the 3x3x3 split model analyses. The mesh is 36 elements in
the x direction by 34 elements in y, and 29 elements in z giving 35,496 total elements
and 116,559 total degrees of freedom.
Figure 5.8: Mesh used for all 3x3x3 split models.
69
5.1.2 Results
The results for the 3x3x3 models will be presented in this section. The material
properties used in the analysis are the same as in Table 4.1 where Ef/Em = 30.
It should be noted that each of the 27 small models were analyzed as RVEs where
appropriate periodic boundary conditions were used for all models. The results are
presented in groups of 9 which share the same z-locations where labels are given in
Figure 5.2.
Material 111 211 311 121 221 321 131 231 331 Properties
vf 0.428 0.419 0.430 0.424 0.321 0.483 0.437 0.398 0.468 E11/Em 4.20 4.45 4.48 3.94 2.77 4.76 4.17 4.29 4.69 E22/Em 7.65 7.09 6.90 8.49 5.85 9.36 7.82 6.64 7.88 E33/Em 3.81 4.03 4.19 3.87 2.66 5.15 3.97 3.41 4.96
ν12 0.10 0.10 0.11 0.09 0.10 0.09 0.10 0.10 0.10 ν21 0.18 0.17 0.17 0.19 0.21 0.18 0.18 0.16 0.17 ν13 0.21 0.21 0.19 0.21 0.25 0.17 0.21 0.23 0.17 ν31 0.19 0.19 0.18 0.20 0.24 0.18 0.20 0.18 0.18 ν23 0.20 0.19 0.18 0.19 0.23 0.17 0.18 0.21 0.16 ν32 0.10 0.11 0.11 0.09 0.11 0.10 0.09 0.11 0.10
G12/Em 1.84 1.87 1.83 1.87 1.33 2.18 1.82 1.73 1.88 G23/Em 1.84 1.81 1.80 1.99 1.35 2.47 1.87 1.57 2.06 G31/Em 1.32 1.50 1.47 1.27 0.95 1.59 1.34 1.32 1.55
Table 5.1: Finite element results for 1st layer in Z-direction of the 3 x 3 x 3 model. (Labels above indicate x, y, and z cube positions respectively, i.e. 111 starts at the origin)
Table 5.1 contains all models having relative z location of 1 (i.e. 221). First note
that the volume fraction of the center bottom model is significantly less than the
others on this level. All other models were above 40% volume fraction and model-221
was only composed of about 32% fibers indicating a resin-rich core within the model.
70
This essentially turns the composite into a complicated box beam. The second item
of interest from the model data is the fact that the composite sample is significantly
stiffer in the y-direction. This is not surprising given that the slice data shows that
a majority of fibers are aligned in this direction.
Material 112 212 312 122 222 322 132 232 332 Properties
vf 0.497 0.410 0.448 0.454 0.351 0.470 0.443 0.410 0.504 E11/Em 5.02 4.26 4.72 4.53 3.02 4.74 4.48 4.53 5.47 E22/Em 10.09 7.54 7.53 9.81 6.86 9.23 7.84 7.39 8.54 E33/Em 4.93 3.84 4.37 3.51 2.95 5.04 4.38 3.52 5.54
ν12 0.08 0.09 0.10 0.08 0.09 0.09 0.10 0.10 0.10 ν21 0.16 0.16 0.16 0.17 0.20 0.18 0.18 0.16 0.16 ν13 0.18 0.21 0.19 0.21 0.24 0.17 0.18 0.22 0.16 ν31 0.17 0.19 0.18 0.18 0.23 0.18 0.18 0.17 0.16 ν23 0.16 0.19 0.17 0.19 0.22 0.17 0.17 0.20 0.16 ν32 0.08 0.10 0.10 0.08 0.10 0.10 0.10 0.10 0.10
G12/Em 2.16 1.81 1.96 2.13 1.49 2.27 1.92 1.90 2.21 G23/Em 2.32 1.77 1.91 2.05 1.52 2.39 2.02 1.66 2.30 G31/Em 1.56 1.39 1.55 1.33 1.01 1.63 1.43 1.34 1.85
Table 5.2: Finite element results for 2nd layer in Z-direction of the 3 x 3 x 3 model. (Labels above indicate x, y, and z cube positions respectively, i.e. 222 is the center cube)
In a similar manner, Table 5.2 contains results from all models having a relative
z location of 2. As in z location 1, the center model for this layer also has a com-
paratively low volume fraction, around 34%. Also, the y-direction E22 is significantly
higher than that in the other two directions. The x-direction seems to be slightly less
compliant than the z-direction, but not in all cases.
Finally, Table 5.3 contains results for all models

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ELASTIC PROPERTY PREDICTION OF LONG FIBER COMPOSITES USING …