FINITE ELEMENT BASED SIMULATION OF PHASED COMPOSITE MATERIAL FOR AIRCRAFT DESIGN by ADRIAN ALEGRE BETH TODD, COMMITTEE CHAIR STEVE SHEPARD STEVE DANIEWICZ A THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mechanical Engineering in the Graduate School of The University of Alabama TUSCALOOSA, ALABAMA 2019
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FINITE ELEMENT BASED SIMULATION
OF PHASED COMPOSITE MATERIAL
FOR AIRCRAFT DESIGN
by
ADRIAN ALEGRE
BETH TODD, COMMITTEE CHAIRSTEVE SHEPARD
STEVE DANIEWICZ
A THESIS
Submitted in partial fulfillment of the requirementsfor the degree of Master of Science
in the Department of Mechanical Engineeringin the Graduate School of
The University of Alabama
TUSCALOOSA, ALABAMA
2019
Copyright Adrian Alegre 2019ALL RIGHTS RESERVED
ABSTRACT
Composite materials are frequently used in the aircraft and aerospace industries for
creating lighter, stronger and cheaper materials. In short, a composite material is a ma-
terial made up of two or more constituent materials that seek to exploit the most advan-
tageous aspects of each material. The design and development process of composite ma-
terials has seen little change in recent years despite an increasing necessity. The current
methodology for designing composites in the aircraft industry, described in Composite Ma-
terials Handbook [37], utilizes the ”Building-Block” approach. In this approach composites
are extensively tested, with tests rising in number and complexity, as the size of the com-
posite elements increases. This method is used due to the inability to predict composite
behaviors and can lead to high cost and time inefficiency.
This work presents an FEM based simulation of composite materials in order to cir-
cumvent large-scale testing by accurately predicting composite behavior. A common air-
craft composite material, Al/SiC, was replicated and verified against empirical data from
literature. Individual material simulations for Al and SiC were first developed and veri-
fied. Two subsequent analyses of the materials combined as a composite were performed in
which the percent weight fraction of SiC was varied. These simulations include Al as the
matrix and SiC as spherical inclusions. Analysis of stress-strain curves for the simulated
composite material demonstrated agreement with empirical data from the literature. This
thesis outlines the processes of geometry development, geometry implementation, simula-
tion set-up, and data analysis of the study.
ii
DEDICATION
This work is dedicated to my brother, Marco Alegre, without whom this work would
have been finished much sooner. Thank you for always being there for me and offering
many opportunities to divert my attention.
iii
LIST OF ABBREVIATIONS AND SYMBOLS
%wt.fr. Percent-Weight-Fraction
ρ Material density
m Body Mass
N Number of Specified Object
V Body Volume
Al Aluminum
Mg Magnesium
SiC Silicon-Carbide
CMC Ceramic Matrix Composite
F Force
FEM Finite Element Method
MD Material Designer
MMC Metal Matrix Composite
PMC Polymer Matrix Composite
RVE Representative Volume Element
SEM Scanning Electron Microscope
SFRC Short Fiber Reinforced Composite
UD Uni-Directional
iv
ACKNOWLEDGMENTS
This study was supported by the National Science Foundation’s (NSF) Louis Stokes
Alliance for Minority Participation (LSAMP) Fellowship. This fellowship assists under-
represented minority groups in obtaining graduate degrees in the STEM field. I would
like to thank Dr. Viola Acoff for accepting me as a recipient of this fellowship and offer-
ing guidance in various aspects of academia to ease the transition to graduate school. I
would also like to thank my committee members Dr. Beth Todd, Dr. Steve Shepard, and
Dr. Steve Daniewicz for their expertise and aid. I am thankful for Dr. Beth Todd’s con-
sistent support and counsel despite her regularly overloaded schedule. I would also like to
express gratitude to Dr. Steve Shepard for promptly addressing any and all concerns re-
lated to the department over the years and to Dr. Steve Daniewicz for his membership at
such short notice with informative feedback. Furthermore, I am appreciative of Dr. Vinu
Unnikrishnan for offering guidance and navigation on building simulation credibility, and
assisting with the basis of this research.
I would also like to take this opportunity to thank my personal mentor Dr. Laurie
Carrillo for offering advice and direction over the years. Without her knowledge and en-
couragement this research would not be possible. Lastly, I want to acknowledge my fel-
low graduate students, David Leech and Miranda Tanouye, for their support and feedback
APPENDIX A Various ANSYS Plots of Modeled Bodies . . . . . . . . . . . . . . . . 53
APPENDIX B MATLAB Script for current and future work . . . . . . . . . . . . . . 58
APPENDIX C JavaScript Script for current and future work . . . . . . . . . . . . . 63
viii
LIST OF TABLES
4.1 Material Parameters for Constitutive Components in MMC . . . . . . . . . . 23
4.2 Relative error between desired weight fraction and actual weight fraction . . 29
ix
LIST OF FIGURES
1.1 Types of composites based on reinforcement shape and composite form (a)Layered Composites (b) Phased Composites . . . . . . . . . . . . . . . . . . 2
1.2 Flowchart representation of the “Building-Block” Approach . . . . . . . . . . 3
3.1 Position vector x and velocity vector v. Body force f dV acting on an ele-ment dV of volume and surface force TdS acting on an element dS of sur-face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 The nine components of a stress tensor, where the first subscript denotesthe normal direction, and the second subscript indicates the direction of thestress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Isotropic hardening model: Demonstration of uniform increase in yield surface 16
3.5 Kinematic hardening model: Demonstration of shift in yield surface . . . . . 17
3.6 Demonstration of dividing a continuous shape into a finite number of ele-ments Number of elements increase with decrease in element size . . . . . . . 19
3.7 Example of mesh convergence where stress data converges to single valueNumber of elements increases from a to e with other parameters constant . . 19
3.8 Illustration of Cubic Element with nodes labeld A through H . . . . . . . . . 20
3.9 Example 1 of contour plot for strains within a complex geometry . . . . . . . 21
3.10 Example 2 of contour plot for strains within a complex geometry . . . . . . . 22
4.1 Representation of (a) Bi-Linear Isotropic Hardening and (b) Multi-LinearIsotropic Hardening correlation to data . . . . . . . . . . . . . . . . . . . . . 24
4.2 Mesh Convergence for simple cube with tetrahedron elements . . . . . . . . . 25
4.3 Representation of SOLID187 element geometry with nodes I through R . . . 26
4.5 Flow chart representation of MATLAB logic for point selection . . . . . . . . 28
4.6 Representative output from MATLAB of Sphere center points in boundingbox Sphere centers indicated in dark blue, bounding box in light blue . . . . 30
4.10 Demonstration of boundary conditions applied to composite geometry . . . . 34
4.11 Demonstration of stress calculations for the individual components . . . . . . 36
5.1 Comparison of empirical data from F. Shehata et al. vs FEM material modelfor Commercially Pure Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Comparison of hand calculations to FEM material model for SiC . . . . . . . 39
5.3 Comparison of empirical data from F. Shehata et al. VS FEM materialmodel for 5%wt.fr. of SiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4 Comparison of empirical data from F. Shehata et al. VS FEM materialmodel for 10%wt.fr. of SiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.5 Side by side comparison of all data F. Shehata et al. VS FEM model data . 41
5.6 Example of required periodicity in ANSYS Material Designer . . . . . . . . . 43
5.7 Example of non-periodic RVE that is not accepted by Material Designer . . 43
5.8 Example of aligned fiber generation for future work . . . . . . . . . . . . . . 44
5.9 Example of randomly oriented fiber generation for future work . . . . . . . . 45
A.1 Example 1 of scalability designed into modeling process with spheres . . . . 53
A.2 Example 2 of scalability designed into modeling process with spheres . . . . 54
A.3 Demonstration of inclusions embedded in bounding box geometry Boundingbox extruded as frozen material . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.4 Example of scalability of number of inclusions and inclusion size . . . . . . . 55
and SBDs (Schottky Barrier Diodes). These transistors have allowed for fast computation
on smaller and smaller computer chips [17]. Silicon-Carbide is also still used as an abrasive
in products such as sandpaper, grip tape, water-jet cutters, and many others [5]. In struc-
tural materials, it is included as a reinforcement to improve overall strength. In the 1990s
Silicon-Carbide was studied as a reinforcing material for turbine blades [40]. This increase
in strength is also utilized for safety equipment, such as bullet-proof vests, that employ
Silicon-Carbide disks [38]. The inclusion of Silicon-Carbide with Aluminum was found to
produce a light but strong composite material, suitable for the Aircraft and Aerospace in-
dustries [57].
2.2 Analysis
Current analyses of Al/SiC composites can be divided into two broad categories: Em-
pirical and Simulation based study. A common analysis of empirical studies is the exam-
ination of the affects of varied %wt.fr. of SiC, or other reinforcement materials, on the
mechanical behavior of the composite [18, 44, 46]. Other studies have included the wear
rate, and machinability of Al/SiC composites for specified %wt.fr. [21, 55]. Increasing the
wettability of SiC with Mg, and enhancing the overall bonding between the matrix and
reinforcement has also been an area if interest [46]. Simulated studies have examined key
principles that are more difficult to examine in empirical testing. A common analysis of
simulated studies is the examination of particle size, and intermediate distance between
particles on the stress behavior of a matrix as a 2D simulation [15, 26, 55]. 2D simula-
6
tions have also been employed for studies on reinforcement debonding (disconnect from
matrix) and crack propagation [28, 55]. A more accurate representation of SiC particles, as
straight-edged polygons, has been studied in 2D to determine peak stresses around parti-
cles [55]. While all of these simulated studies offer insight into the mechanical behavior of
Al/SiC composites, a common aspect is the simplification of composite geometries into a
2D plane-strain simulation [15, 21, 26, 55]. This study focused on developing a 3D simula-
tion of an Al/SiC composite. Data from F. Shehata et al [44] was utilized to verify simula-
tion results. This study included two simulations of varying SiC %wt.fr., based on testing
from [44]. Further details on the simulation parameters are discussed in chapter 4.3. Re-
sults of the simulation can be seen in chapter 5, and future study following this work is
discussed in 5.4. The overall purpose of this thesis was to develop a unique simulation to
be used in the design process of Al/SiC composites in the Aircraft and Aerospace indus-
tries.
7
CHAPTER 3
MATHEMATICAL BACKGROUND
3.1 Basic Principles
For solutions in either solid or continuum mechanics, there are 3 major considerations
that must be taken into account:
• Newtonian Equations of Motion
• Geometry of Deformation
• Stress-Strain Relations
The Newtonian equations of motion describe the principles of conservation of linear
and angular momentum for finite bodies (as opposed to a single point) and the associ-
ated concept of stress within those bodies. The geometry of deformation is the formula-
tion of strain with regard to the gradients of a displacement field for those bodies. Lastly,
the stress-strain relations help to characterize macroscopic behavior of the material under
analysis. These three principles are adequate to describe most mechanics solutions. Other
principles not discussed here included relations of diffusion of constituent materials, heat
and energy considerations, and others associated with electrical properties. Other neces-
sary formulation and considerations are also discussed in this chapter.
8
3.2 Newtonian Equations of Motion
3.2.1 Linear and Angular Momentum
Let x denote a position vector of a point in space relative to the origin of the refer-
ence frame with the components (x1, x2, x3) as shown in figure 3.1. Suppose a material
occupies the space illustrated in grey, where v = v(x, t) is the velocity vector of this mate-
rial at position x for time t. After an infinitesimal interval of dt, the material point will be
positioned at x + vdt.
Figure 3.1: Position vector x and velocity vector v. Body force f dV acting on anelement dV of volume and surface force TdS acting on an element dS of surface
At the basis of the Newtonian Equations of Motion are the principles of linear and
angular momentum such that:
P =
∫V
ρvdV (3.1)
and
H =
∫V
ρx× vdV, (3.2)
Where P and H denote linear and angular momentum respectively [39]. These integrals
9
describe the linear and angular momentum for a body that occupies a volume V at a point
in space x (in Cartesian coordinates) where ρ = ρ(x, t) is the mass density of the volume
at position x with respect to time. We can then describe the force and moment acting on
a body as:
F =∂P
∂t(3.3)
and
M =∂H
∂t. (3.4)
When dealing with bodies at static equilibrium, under the principle of conservation of mo-
mentum F = 0 and M = 0. This allows for the calculation of P and H for a given time
[39].
3.2.2 Stress
If F and M are assumed to be body forces f, defined as f dV acting on a volume ele-
ment of dV, and surface forces represented as a stress vector T, with TdS defined as sur-
face force acting over a surface element dS, as shown in figure 3.1, then linear and angular
momentum can be written as:
∫S
TdS +
∫V
fdV = F =∂P
∂t=
∫V
ρadV (3.5)
and ∫S
x×TdS +
∫V
x× fdV = M =∂H
∂t=
∫V
ρx× adV, (3.6)
Where a denotes acceleration as a = a(x, t). This leads to the classification of the stress
components, σij, which are nine quantities that vary with position and time, and act on
the surface of a body. For a given point x where a surface normal is oriented in the posi-
tive xi direction (i = 1,2 or 3 for 3D coordinates), then σi1,σi2, and σi3 are Cartesian com-
ponents of the stress vector T. Thus the stress σij is the stress in the j direction with re-
spect to the i face of the body.
10
Figure 3.2: The nine components of a stress tensor, where the first subscript denotesthe normal direction, and the second subscript indicates the direction of the stress.
These nine quantities make up the components of a 3 x 3 matrix that is the stress ten-
sor, T , such that:
T =
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
(3.7)
This matrix can be translated and transposed with relation to the original matrix, where
multiplication is such that if [A] = [B][C] then Aij = Bi1C1j +Bi2C2j +B13C3j [39].
3.2.3 Equations of Motion
Taking the multiplication properties of the stress tensor into consideration, we can
then define a second order tensor such that [σ′] = [a][σ][a]T . Using this expression in com-
bination with the divergence theorem:
∂σ1j∂x1
+∂σ2j∂x2
+∂σ3j∂x3
+ fj = ρaj (3.8)
11
where j = 1, 2, and 3. These equations are applicable when σij values are continuous and
differentiable, thus describing motions for a continuum. For further simplification, σij =
σji (i,j = 1, 2, and 3) as derived from angular momentum principles, causing the stress
tensor to be symmetric [39].
3.2.4 Principal Stress
The symmetry of the stress tensor leads to the crucial concept that for any point in
space, x, there exists a set of three mutually orthogonal directions where there is no shear
stress. These directions are referred to as the principle stress directions, and stresses along
these directions as the principle stresses. These stresses are the eigenvalues, s, while the
principle stress directions are the eigenvectors, n, such that: T = sn. A solution to this
eigenvalue problem is such that:
([σ]− s[I]) = −s3 + I1s2 + I2s+ I3 = 0 (3.9)
where:
I1 = tr[σ] (3.10)
I2 =1
2(σσ − I21 ) (3.11)
and
I3 = det[σ] (3.12)
Where tr is indicative of the trace, or sum, of diagonal elements in a matrix, and det is the
determinant of that [39].
12
3.3 Geometry of Deformation
3.3.1 Strain and Strain-Displacement Relations
The deformation of a solid structure implies its change in shape over a period of time.
This deformation, referred to as strain, is typically measured with respect to some refer-
ence configuration of the solid that is classified as an undeformed state. If the time mea-
sured when the reference configuration for a solid begins to exist is set to be zero, and
X describes the position vectors for that solid, then deformation may be described as
x = x(X, t) and it is known that x = x(X, 0) = X . Displacement can then be de-
scribed as a vector, u from the initial position to some arbitrary position at time t such
that u = x (X,t) - X. The extension for velocity and acceleration can then be made such
that v = ∂x(X, t)/∂t and a = ∂2x(X, t)/∂t2. These equations are applicable when the
reference configuration deformation is sufficiently small such that |∂ui/∂Xj| << 1. These
conditions are typically met in practice when examining sufficiently rigid materials that
elicit a linearized expression for strain [39].
3.3.2 Small Strain Tensor
For sufficiently small strains, εij, where |∂uk/∂Xl| << 1 for all k and l values, a set of
material strains can be defined as:
εij =1
2(∂uj∂Xi
+∂ui∂Xj
) (3.13)
where i, j = 1, 2, 3. Similar to the stress components, strain components are symmetric
such that εij = εji given the existance of principle strain directions (i.e. there exist three
mutually perpendicular directions at a given point). Despite this symmetry, it is impor-
tant to note that strains do not vary arbitrarily between various points in the body. This
is due to the fact the the six strain components are derived from three displacement com-
ponents [39].
13
3.3.3 Finite Deformation and Strain Tensor
Finite deformation theory is applicable to all bodies regardless of size. For a small
body deformed from initial vector dX to final vector dx during an arbitrary time t the de-
formation gradient is define as Fij = ∂xi(X, t)/∂Xj. The components of F can be describe
by a symmetric matrix [U ] and rigid rotation matrix [R] via the polar decomposition the-
ory such that: [F ] = [U ][R]. For example, a set of three fibers mutually orthogonal to one
another that have undergone stress in the principle direction are said to have undergone
extension strain without shear strain between them. In this case the deformed fibers are
said to have remained orthogonal but rotated by some operation [R] [39].
3.4 Stress-Strain Relations
3.4.1 Linear Elasticity
If a material is stressed within its linear elastic deformation region, then the stresses
and strains of that material are proportional to one another and scaled by a factor, E,
called Young’s Modulus. This relationship, known as Hooke’s law, is such that:
σ = Eε (3.14)
For a simple linear elastic bar loaded axially in a principle direction, the resulting strain
can be written as: ε11 = σ/E and ε22 = ε33 = −vε11 where ε12 = ε23 = ε31 = 0 due to
the loading being uniaxial in nature. The parameter v is a unit-less ratio of lateral strain
to axial strain called Poisson’s ratio that is often used to characterize behavior of various
types of materials. These equations most generally apply to isotropic solids that exhibit
the same material response regardless of the direction of stressing. Brittle materials, like
SiC, will only behave in the linear elastic region and tend to have a large Young’s Modu-
lus. This means that the material will exhibit large amounts of stress for small strains, as
shown in figure 3.3 [25].
14
3.4.2 Plasticity
Plasticity describes the deformation of a material beyond the region of a linear elas-
tic response in which the relationship between stress and strain is no longer linearly cor-
related. In a physical sense, the plastic region of a stress-strain curve for a given mate-
rial is indicative of when a material has undergone non-reversible deformation. The point
at which this occurs is referred to as the yield strength of a material. The plastic region
will have various characteristics dependent on material parameters and must be considered
when dealing with deformations expected to exceed the linear elastic region. This behavior
is typically observed in ductile materials, such as Al. For the purposes of this study, plas-
ticity is assumed to be rate-independent such that the strain rate of a material does not
effect the plastic response. If a material is stressed such that it reaches the plastic region it
is said to have undergone hardening as there is a reduction in the change of stress per unit
strain.
Figure 3.3: Summary of Stress-Strain Relations
In order to simulate this behavior there are two categories of hardening: Isotropic and
Kinematic hardening. The primary difference between these two hardening models, is the
way that they affect a material’s yield surface. In terms of mathematics, a yield surface
15
is described by a function called the yield function, F , where F < 0 is within the linear
elastic deformation region and F = 0 is in the plastic deformation region [25]. The most
utilized yield criteria is the von-Mises criteria that states:
When plotted this function creates a plot known as the yield surface, where all points
within this surface are said to be in the elastic region. The primary difference between
isotropic and kinematic hardening is with respect to this surface. For isotropic harden-
ing, this yield surface exhibits uniform expansion in all directions when the yield strength
is reached. Comparitively, for kinematic hardening the yield surface remains unchanged
in sized, but shifts in position. A graphic representation for 2D hardening can be seen in
figure 3.4 and 3.5. For the purpose of this study an isotropic hardening model is used as it
is simple and adept for measuring a monotonic material response. For cyclic loading, both
the number of cycles and type of material must be taken into consideration when deter-
mining whether to use an isotropic or kinematic hardening model [25].
Figure 3.4: Isotropic hardening model: Demonstration of uniform increase in yieldsurface
16
Figure 3.5: Kinematic hardening model: Demonstration of shift in yield surface
3.5 Rule of Mixtures
Consider a composite of matrix material m with embedded reinforcements r. As dis-
cussed in 3.4.1 it can be assumed that σm = Emεm and that σr = Erεr. If the average
stress of the composite, σc, acts across the cross-sectional surface area of the Representa-
tive Volume Element (RVE), A, then it follows that:
σcA = σmAm + σrAr (3.16)
Taking the stress-strain relationship into consideration for the overall Young’s Modulus of
the composite, Ec it can then be said that:
Ec = ErAr
A+ Em
Am
A(3.17)
simplified to
Ec = ErVr + EmVr (3.18)
since the volume fraction of reinforcement can be written as Vr = Ar
Aas well as for the
matrix [31]. This general methodology is known as the Rule of Mixtures. This concept is
17
then extended such that:
σc = σrVr + σmVr (3.19)
and
εc = εrVr + εmVr (3.20)
These equations allow for calculation of a composite material’s predicted stress and strain
based on the volume fraction of its constituent materials. This methodology was utilized
to process data from the FEM analysis.
3.6 Finite Element Method
An engineer designing a mechanical structure will need to know how the structure
will behave under load. The equations deriving structural stresses are known, but they
cannot be directly solved for a complicated geometry such as a bridge, tower, walkway,
etc. However, these equations can be solved for very simple shapes. The finite element
method replaces a complex geometry with an approximately equivalent network of sim-
ple elements, as shown in figure 3.6. The overall pattern of elements is known as the finite
element mesh, and is a pattern unique to each engineering problem. The initial step is to
design this mesh by choosing how many, and what kind of elements to use. The accuracy
of the calculation depends on the number of elements in the mesh. As the number of ele-
ments increases, the average size of the elements decreases. The smaller the elements are
the more accurate the calculations will be and the more accurate the results. However,
more elements require more calculations. For efficiency purposes, just enough elements are
chosen to give an adequate accuracy while also ensuring a reasonable computation time.
This process, known as mesh convergence, compares data values of different mesh sizes in
order to determine where these values plateau and are no longer dependent on mesh size
(figure 3.7).
18
Figure 3.6: Demonstration of dividing a continuous shape into a finite number ofelements
Number of elements increase with decrease in element size
Figure 3.7: Example of mesh convergence where stress data converges to single valueNumber of elements increases from a to e with other parameters constant
Choosing the shape of the elements is also crucial for accuracy. The example in fig-
ure 3.8 illustrates 8 points, called nodes, which define a simple cubic element where there
is one node at each vertex of the cube. In general, these nodes have a total of 6 degrees of
freedom: translations and rotations in X, Y, and Z. The exceptions would be the nodes to-
wards the outside edge of the overall geometry that must abide by a given boundary con-
dition. These boundary conditions must be included to complete the description of the
physical problem so that the solution will be uniquely defined. Finally, the elastic proper-
ties of the simulated materials are specified as well as specific boundary conditions (load-
ing, fixed supports, rotations, etc.).
19
Figure 3.8: Illustration of Cubic Element with nodes labeld A through H
The mathematical analysis aims at deriving an equation to describe the whole sys-
tem. The analysis begins by expressing the displacement of any node as a function of its
coordinates in X, Y, and Z. In the above example there are 8 nodes in the element, giving
a total of 8 equations of displacement. These 8 equations form a matrix, that is now the
starting point for a series of steps based on the fundamental laws of mechanics. The initial
step relates displacements to stresses. From these stresses the strain energy is obtained,
followed by the derivation of potential energy. Finally, from minimum potential energy a
set of system equations is determined for the complete element. This new matrix is called
the stiffness matrix for the element. Instead of a single displacement, x, the matrix oper-
ates on the vector x whose components are the displacements for the whole element. This
same process is carried out for each element in the mesh.
The next step is to combine these matrices into a single, large matrix, representing
the stiffness of the whole system. Neighboring elements will have nodes in common, so val-
ues of these nodes will occur in matrices for both elements. The matrices are thus com-
bined using a simple merging technique. This process is done concurrently with the pro-
cess of solving for the overall system equation. This whole process is known as reduction.
As mentioned previously, the components of the matrices represent a set of simultaneous
equations. To reduce the size of the matrices, the first equation is solved and the solution
20
is substituted into subsequent equations.
This process repeats until the matrix for the next element is included to be solved.
When all of the element matrices have been solved, the resulting solution is for a single
node. This result acts as a key, working backwards through the equations of the system
until the displacement of every node in the system is obtained. From these results the cor-
responding stresses, strains, and other data can be easily calculated and represented in a
contour diagram, where intermediate values between nodes are calculated via interpola-
tion. An example can be seen in figures 3.9 and 3.10, where areas in blue represent low
strain and areas in red represent high strain.
Figure 3.9: Example 1 of contour plot for strains within a complex geometry
21
Figure 3.10: Example 2 of contour plot for strains within a complex geometry
22
CHAPTER 4
SIMULATION
4.1 Material Parameters
As discussed in chapter 2.2, data from [44] was utilized to develop an Al/SiC com-
posite simulation. In order to develop this simulation two individual sub-simulations for
Al and SiC were created. These simulations were then matched to experimental data and
theoretical calculations for the Al and SiC respectively, to verify accuracy of the mate-
rial replication. As described in [44], commercially pure Al is typically composed of 99%
Al, with small percentages of various other metals including Ti, Zn, Ni, Mg, Mn, Cu, Fe,
and Si. For the purposes of this study, these small percentages are not taken into consid-
eration when forming the base material as only the macroscopic response is needed. The
FEM software used for this study allows for user defined materials based on various input
parameters. The following parameters were utilized for material generation:
Figure 4.7: Finished composite geometry following automated procedure
31
Figure 4.8: External planes used to cut away excess materialPlanes are scaled by box dimensions.
4.4 Simulation Set Up
4.4.1 Assumptions
In order to simulate the composite material some boundary condition assumptions
were implemented in the set up. The first assumption was that the constitutive compo-
nents of composite were perfectly bonded. To simulate this bond, the contact pairs be-
tween the spherical inclusions and the matrix were assigned to a bonded-contact constraint.
This type of constraint uses a Pure-Penalty calculation method, in which a finite contact
force, Fn, is a function of normal contact stiffness, kn and penetration distance,xp, such
that Fn = kn × xp. The higher the contact stiffness, the lower the amount of penetra-
tion. Ideally, for an infinite kn the penetration would be equal to zero. However, this is
not numerically possible. As long as the penetration was reasonably small, it was consid-
ered negligible and results retained accuracy. This type of contact was applied to the outer
surfaces of the spheres and the corresponding contact surfaces inside the matrix.
32
Figure 4.9: Representation of Pure-Penalty calculation
Secondly, for Al/SiC composites it is understood that the SiC reinforcements typically
vary in size and geometry, with straight edges and a relatively low aspect ratio. Realistic
particles can have complex polygonal geometries, with a wide variety of features. In order
to save on computational time and effort, inclusions were assumed to be spherical with
identical radii, as previously mentioned at the beginning of, and throughout this chapter.
The final assumption was the size of the RVE. In order to improve computational effi-
ciency, the RVE was reduced to have the smallest volume possible while still maintaining
the desired reinforcement volume fraction. Based on SEM imaging, a model with a bound-
ing box of 400 µm x 400 µm x 400 µm was determined to be sufficient and then reduced to
18
in size at 200 µm x 200 µm x 200 µm. The %wt.fr. was also scaled accordingly, resulting
in fewer spheres required to meet the desired SiC content.
4.4.2 Loading and Constraints
Testing in [44] placed finalized composite samples under uniaxial tension and recorded
their stress-strain response. Replicating this procedure, the finalized geometry was placed
under tension using two basic boundary conditions. First, a fixed support was placed at
the base of the bounding box geometry, parallel to the ZX-plane. This condition prevents
translation and rotation in all directions. This condition was used primarily for its sim-
plicity and a broad application to all degrees of freedom. Secondly, a force was applied to
the top surface of the bounding box geometry, normal to the surface, in the positive Y-
direction. For the material verification of commercially pure Al, a force of 1.5× 106µN was
33
applied. This value was chosen to produce a desired stress of 37.5 MPa, primarily to ex-
tend the Al material model beyond the linear elastic region (yield strength = 35 MPa,
table 4.1) and into the plastic deformation. This allowed for the demonstration of accu-
racy of the Al material model. Following the addition of spherical inclusions, the normal
force was reduced to values between 1.0× 106µN and 1.2× 106µN for better convergence.
As discussed in chapter 4.3.2, excess sphere volumes were cut away from the final model.
This resulted in various, circular cross-sectional faces at the boundaries where support
and loading conditions were applied. In order to reach simulation convergence successfully,
these faces were also included as part of the geometry selection on which the boundary
conditions were applied. Application of boundary conditions can be seen in figure 4.10.
(a) Force applied to top surface ofcomposite
geometry in the +Y-direction(b) Fixed support applied to bottom
surface of geometry parallel to ZX-plane
Figure 4.10: Demonstration of boundary conditions applied to composite geometry
4.4.3 Analysis Settings and Output
For this study a single load step was utilized with an end time of 1 second. As dis-
cussed in chapter 3.4.2, the plasticity for Al was determined to be rate independent, and
therefor a 1 second load step was sufficient. The analysis was broken down into sub-steps,
automated by the software, with the following conditions: initial number of sub-steps =
34
10, minimum number of sub-steps = 10, maximum number of sub-steps = 100. A sub-
step is a point within a load step in which a solution is calculated. This approach was
utilized as it provides more accurate results for non-linear, static analyses by gradually
applying loading and boundary conditions. This type of automation also allows the soft-
ware to choose where and when in the loading process to include more or fewer sub-steps,
as needed. By specifying the minimum number of sub-steps, no fewer than 10 data points
were output for the entire load step. When performing FEM analysis, it is crucial to ad-
equately determine how many sub-steps will be necessary for a simulation process as too
few steps can lead to inaccurate data and too many steps can cause convergence failure.
As a general method, it is suggested to begin with fewer sub-steps and slowly increase the
total number steps until the desired amount of data points is achieved. A direct solver was
chosen to perform the mathematical analysis of this system. A direct solver, as opposed to
an iterative solver, is based on the direct elimination of equations by factorization of ma-
trices. This type of solver is typically used for robustness and solving speed when dealing
with non-linear analysis, or for linear analysis with poorly shaped elements [43]. Lastly,
large deflection affects were taken into consideration. By including these affects the solver
was able to account for changes in material stiffness that were influenced by a change in
material shape. As a general rule, it is suggested that these affects are utilized unless it is
known that large deflection will not occur in the simulation environment [42]. Output data
for the simulation was in the form of stress and strain for the individual components of the
matrix and the reinforcements, where reinforcement data represented the whole group as
shown in figure 4.11. The stress and strains recorded were the Equivalent Total Strain and
Equivalent von-Mises Stress. This data was extracted into excel and further processed as
described in the results section.
35
(a) Matrix (b) Spherical Reinforcements
Figure 4.11: Demonstration of stress calculations for the individual components
36
CHAPTER 5
RESULTS AND DISCUSSION
The equivalent total strain and von-Mises stress were output by the FEA software
for the spherical reinforcements as a group and the matrix bounding box. These results
were then exported to excel where final values of stress and strain were calculated using
the Rule of Mixtures for each data point.
5.1 Results
Figure 5.1 illustrates the correlation between the aluminum material model and the
empirical data derived from F. Shehata et al [44]. Despite error near the yield point, the
overall data is in good agreement with the experimental data. However, this correlation
was expected due to the fact that this model utilized the empirical data as input argu-
ments to generate the plastic region. Figure 5.2 presents the comparison between hand
calculations and the FEM material model for SiC. These calculations were performed us-
ing equation (3.14) as a simple approach to model verification. As expected, the FEM ma-
terial model demonstrated good agreement with the calculated values for stress and strain.
Figure 5.3 demonstrates the agreement between the empirical data and FEM model for
the 5%wt.fr. geometry. While the Young’s Modulus was determined to be slightly lower,
and the ultimate strength slightly higher, the general trend of the FEM model matched
that of the experiment. It is speculated that these deviations in data may be due to a
lower number of spherical inclusions for the 5%wt.fr. geometry. This may have led to an
anisotropic response from the material, given the random dispersion of inclusions. How-
37
Figure 5.1: Comparison of empirical data from F. Shehata et al. vsFEM material model for Commercially Pure Al
ever, this hypothesis was not examined in further detail following the conclusion of the
study. The error between the curves measured 9.78% at its maximum and 4.2% on aver-
age. These values were derived from stress-strain output data of the reinforcements and
matrix separately, and then calculated using a weighted average following the form of
equations 3.19 and 3.20.
Figure 5.4 illustrates the correlation between empirical and simulation data for 10%wt.fr.
geometry. Similar to the previous geometry, the higher accuracy of this model may be
attributed to an increased number of spherical inclusions. This may have lead to a re-
sponse more reflective of an isotropic material. However, this hypothesis was also not stud-
ied further following the conclusion of the study. These data were also calculated using a
weighted averaged of individual stress-strain data points derived from the particles and
matrix, separately. The error between the FEM simulation and empirical data was mea-
sured to be 4.55% at its maximum, and 2.1% on average.
Figure 5.5 was included to present the data from the commercially pure Al, 5%wt.fr.,
38
Figure 5.2: Comparison of hand calculations to FEM material model for SiC
and 10% wt.fr. along side one another. Simulations were constrained to 5% and 10%wt.fr.
as these were the only two cases studied in F. Shehata et al [44]. However, given the gen-
eral trends of the data, this model may be able to accurately predict behaviors of Al/SiC
MMC with higher percentages of SiC. In the following sections, improvements and alterna-
tive approaches to the simulation of composite materials is discussed.
39
Figure 5.3: Comparison of empirical data from F. Shehata et al. VSFEM material model for 5%wt.fr. of SiC
Figure 5.4: Comparison of empirical data from F. Shehata et al. VSFEM material model for 10%wt.fr. of SiC
40
Figure 5.5: Side by side comparison of all dataF. Shehata et al. VS FEM model data
5.2 Improvements
The MATLAB script utilized for creating geometry is written such that the total num-
ber of spherical inclusions for a large volume fraction can be specified but are not guaran-
teed. This can be an issue if a known or desired large volume fraction is sought after for
study. Despite this aspect of the scripted logic, this method worked well in simulating the
exact number of reinforcements for relatively small volume fractions of less than 25%. Fur-
ther changes to this script would primarily address this volume fraction issue to increase
its applicability to a wider variety of composite simulations.
Secondly, the automation of geometry drawing by JavaScript required user input that
may lead to error. As stated above, the number of data points output from the MAT-
LAB portion can be unknown and will likely vary between consecutive script runs. This
requires the manual update of the JavaScript script following each run of the program.
Similarly, the bounding box geometry parameters and desired spherical radius must also
be updated if changed in the MATLAB script. Despite the fact that variables for these
41
values are included at the beginning of the JavaScript script, changing these variables may
be confusing if the user is unfamiliar with how to read and understand code. For this rea-
son, future iterations would likely include logic to extract this information from MATLAB
to JavaScript and reduce user input.
5.3 Alternative Methods
The FEM software utilized for this study, ANSYS, includes an analysis system de-
signed to homogenize the material response of RVE constituents for later use in subse-
quent analysis systems. This system, named Material Designer (MD), allows the user to
choose from predefined RVE geometries, including various lattice structures, UD aligned
fibers, UD aligned fibers with varying angles of inclination, randomly oriented short fibers,
woven fibers for laminae, and the ability to develop a user defined RVE. While this is an
innovative and unique tool, there are limitations to the MD system that ultimately led
to seeking other methods of simulation. Despite the fact that MD allows the user to de-
termine the aspect ratio, fiber radius, and volume fraction for fibers in the matrix, the
fibers themselves are ultimately not random. In order to satisfy the homogenization anal-
ysis conditions, the RVE must demonstrate periodicity at all boundaries. This results in
mirrored patterns of fibers, as shown in figure A.4. This condition also applies to user
defined RVE geometries, making it difficult to model truly random PMCs if a reinforce-
ment is drawn at the boundaries of the RVE ( see figure 5.7). The second caveat of the
MD system is the inability to model hyperelastic materials. While the MD system is able
to generate an averaged material response for constituent materials, it is ultimately only
able to do so for linearly elastic materials. This, unfortunately, excludes the MD system
from use in contemporary research that is primarily focused on PMC modeling with elas-
tomeric matrices. Lastly, the inconsistency in geometry sizing of MD makes it difficult for
the user to fit the system to their needs. The predefined RVE geometries are restricted
to the nano- and micro-scale, where as the user defined RVEs are limited to millimeters,
42
centimeters, and meters. For these reasons the approach to model the RVE in an external
software, MATLAB, was preferred.
(a) View of zx-plane from top face ofRVE
(b) View of zx-plane from bottom face ofRVE
Figure 5.6: Example of required periodicity in ANSYS Material Designer
Figure 5.7: Example of non-periodic RVE that is not accepted by Material Designer
43
Figure 5.8: Example of aligned fiber generation for future work
5.4 Future Work
Future work related to this thesis would include improving the fidelity between the
simulation and experimental data, and extending the study to different material and rein-
forcement types. Reducing the error of simulation results can likely be achieved via alter-
native mathematical methods of material homogenization. Production of simulated com-
posite materials may also be examined such that empirical data is derived from in-house
experiments and is easier to acquire. Other reinforcement types that were under consid-
eration, that may be utilized in future studies, include UD aligned fibers and randomly
oriented fibers for SFRC. An altered version of the MATLAB center point selection script,
designed for fibers, is included in B.2. This script similarly selects fibers in a bounding box
geometry under a minimum distance criteria. Intersections between fibers are mathemati-
cally determined using distance calculations. Only fibers without intersections are included
in a master list that is exported for further modeling. Examples can be seen in A.2.
44
Figure 5.9: Example of randomly oriented fiber generation for future work
45
CHAPTER 6
CONCLUSION
In conclusion, this thesis simulated the composite material, Al/SiC, that is typically
utilized in the Aircraft and Aerospace industries. This simulation was verified using empir-
ical data from a published study, with satisfactory agreement between FEA results and ex-
perimental results. While there is room for improvement, the methodology discussed here
offers a simplified approach to composite material simulation. Further work with respect
to this study would include reinforcements with different geometries, as well as different
materials for both the matrix and reinforcements. Composite materials offer several advan-
tages compared to their monolithic counter parts, but require extensive testing throughout
the design process. Composite simulations, such as the one presented in this research, can
be modified and altered to meet the needs of the user, reducing the time and cost associ-
ated with traditional design methods such as the “Building-Block” approach.
46
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52
APPENDIX A
VARIOUS ANSYS PLOTS OF MODELED BODIES
A.1 Current Study
Figure A.1: Example 1 of scalability designed into modeling process with spheres
53
Figure A.2: Example 2 of scalability designed into modeling process with spheres
Figure A.3: Demonstration of inclusions embedded in bounding box geometryBounding box extruded as frozen material
54
(a) Matrix (b) Inclusions
Figure A.4: Example of scalability of number of inclusions and inclusion size
A.2 Future Study
Figure A.5: Demonstration of fiber modeling in MATLAB
55
Figure A.6: Left: All possible fibers in the given volumeRight: Fibers that meet the non-intersection criteria
Figure A.7: Demonstration of generating a 1-to-1 model from MATLAB to ANSYS
56
Figure A.8: Example 1 of scalability designed into modeling process with fibers
Figure A.9: Example 2 of scalability designed into modeling process with fibers
57
APPENDIX B
MATLAB SCRIPT FOR CURRENT AND FUTURE WORK
B.1 MATLAB Point Selection
% This is a random Sphere generator designed to generate spheres within a
% given bounding box. The spheres are conditioned such that they do not
% intersect but may in fact touch on edge. The spheres are plotted inside a
% Cube using the plotcube.m script as a representation. While the number
% of spheres may be specified, only the spheres that meet the minimum distance
% criteria will be kept. The output is a Text File that can be read by
% DesignModeler. Units are consistent throughout script.