Finite Element Study in the Deflection of Composite Laminate Plates
and Functionally Graded Material Plates
by
Rigels Bejleri
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
Master of Engineering
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2014
(For Graduation May, 2015)
i
© Copyright 2014
by
Rigels Bejleri
All Rights Reserved
ii
CONTENTS
Finite Element Study in the Deflection of Composite Laminate Plates and Functionally
Graded Material Plates ................................................................................................. i
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
GLOSSARY .....................................................................Error! Bookmark not defined.
KEYWORDS ................................................................................................................... vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Problem Description........................................................................................... 3
2. Methodology ................................................................................................................ 4
2.1
Methodology Overview ..................................................................................... 4
2.2
Laminated Ply Properties ................................................................................... 4
2.3
Laminated Plate Modeling ................................................................................. 6
2.4
Properties of FGM Plate Elements..................................................................... 9
2.5
Properties of FGM Plate & Mori-Tanaka Method ........................................... 10
2.6
Functionally Graded Material Plate Modeling ................................................. 14
3. Results and Discussion .............................................................................................. 15
3.1
Laminated Plate Results and Validation of Modeling Method ........................ 15
3.2
Functionally Graded Plate Results ................................................................... 17
4. Conclusion ................................................................................................................. 23
References........................................................................................................................ 25
Appendix A...................................................................................................................... 26
iii
LIST OF TABLES
Table 1: Properties of AS/3501 Composite Ply................................................................. 6
Table 2: Young’s Modulus Comparison Between Two Sources....................................... 9
Table 3: Properties of FGM Plate Substrates .................................................................. 10
Table 4: Summary of Plates Studied ............................................................................... 14
Table 5: Summary of FGM Plate Graphite Distribution and Deflections ....................... 19
iv
LIST OF FIGURES
Figure 1: Laminated Composite Plate [4] .......................................................................... 2
Figure 2: Functionally Graded Material [5]....................................................................... 2
Figure 3: CLP Plate Layer (Ply) Arrangement .................Error! Bookmark not defined.
Figure 4: Meshed Sheet Body Representing CLP Plate .................................................... 8
Figure 5: Deflection of 10 Layer Cross Ply Laminated Plate .......................................... 15
Figure 6: Deflection of 20 Layer Cross Ply Laminated Plate .......................................... 16
Figure 7: Deflection of 10 Layer Angle Ply Laminated Plate ......................................... 17
Figure 8: Through Thickness Material Fractions of Plate 8 ............................................ 20
Figure 9: Through Thickness Material Fractions of Plate 11 .......................................... 20
Figure 10: Through Thickness Material Fractions of Plate 1 .......................................... 21
Figure 11: Through Thickness Material Fractions of Plate 5 .......................................... 21
Figure 12: Through Thickness Material Fractions of Plate 6 .......................................... 22
Figure 13: Through Thickness Material Fractions of Plate 7 .......................................... 22
v
LIST OF SYMBOLS
Symbol Unit Meaning
Pascale – unit used to measure pressure and elastic modulus
Pa
E
Pa
ν
G
Elastic Modulus
Poisson’s Ratio
Pa
Q
Shear Modulus
Reduced Stiffness Matrix (Element)
tply
m
Thickness of a laminate ply
Vf
%
Volume Fraction (of fiber reinforcement)
H
m
Thickness or height of the plate
vi
KEYWORDS
Composite Laminated Polymer (CLP)
Functionally Graded Material (FGM)
Finite Element Method
Classical Lamination Theory
vii
ACKNOWLEDGMENT
Type the text of your acknowledgment here.
viii
ABSTRACT
This project studied the deflection of Composite Laminate Plates and Functionally
Graded Material Plates in order to determine which configuration is stiffer. The deflection
of these plates was calculated via the Finite Element Method using the Finite Element
Analysis software ANSYS. The study samples were simply supported square plates with
a uniform pressure applied. Both types of composites were comprised of a set material
pair and a set volumetric ratio of the material pair.
The goal of this study was to gain an understanding of the theoretical deflection
differences between the two types of composites. However, this study did not account for
debits associated with material defects such as voids, gaps, and porosity that could be
present due to the manufacturing processes used to create these composites. Therefore, it
is recommended that extensive tests be conducted to gain a full understanding of the
material capabilities of the plates studied in this project.
ix
1. Introduction
1.1 Background
Humans have taken advantage of the properties of composite materials for
thousands of years. Concrete, plywood, and cob (mud and straw) bricks are some of the
earliest examples of man-made composites that are still used today for construction
throughout the world [1]. Over the past century Fiber Reinforced Plastics (FCPs) gained
popularity in the aerospace industry as an alternative to metal because of their high
strength to weight ratio [1]. Composite Laminated Polymers (CLPs), shown in figure 1,
are a type of FCP that is manufactured by stacking layers of high strength fiber reinforced
polymers in different arrangements to provide required engineering properties [2]. The
layered composition of CLPs provides flexibility in construction and allows for the
creation of a material with near metal, or in some specific aspects, better than metal
properties with a fraction of the weight. Extensive research and development in CLPs has
led to a decrease in the cost of design and manufacture of CLPs, allowing for their broader
use in automobiles, trains, electronics, sports, construction, and many other industries.
Functionally Graded Materials (FGMs) are another type of composite material.
FGMs are composed of a continuously varying and smooth distribution of the two or more
constituents it is made of. FGMs provide an alternative to materials that are used in
applications that require the base material to be coated due to the severe environment that
it operates in. Inter-laminar stresses between the substrate and the coating can build up
overtime due to the adverse operating conditions and relatively abrupt transition of
material properties from the coating to the substrate. These stresses can cause coating loss,
thus exposing the substrate to the potentially corrosive environment. FGMs are a good
alternative in these applications because the smooth transition from coating material
properties to substrate properties reduces the chance for inter-laminar stress build up and
thus reduces the chance for coating loss [3].
CLPs can experience similar de-lamination due to inter-laminar stresses between
the matrix and reinforcing fibers when exposed to a challenging environment such as a
moist environment. This project will study the deflection capability of FGM polymer
plates manufactured via 3D printing and compare it to the deflection of CLP plates
1
composed of the same two materials to see if the FGM plates could be a comparable
alternative to CLPs.
Figure 1: Laminated Composite Plate [4]
Figure 2: Functionally Graded Material [5]
2
1.2 Problem Description
This project will utilize ANSYS to study the deflection behavior of simply
supported Composite Laminate Polymer (CLP) plates and Functionally Graded Material
(FGPs) plates when a uniform pressure load is applied. The goal is to determine which
type of plate, meaning CLP or FGM, has the highest stiffness. In order to isolate the effects
of the composite type on the deflection, all the plates will have the same physical
dimensions, meaning length, width and thickness. In addition, all of the plates will be
composed of the same material couple, an epoxy matrix and graphite reinforcing material.
Lastly the volumetric ratio for the two materials will be held constant for all plates. One
of the CLP plates to be studied will be a cross ply configuration where the plies are stacked
90 degrees relative to each other. The other CLP plate will be an angle ply with plies
stacked at 45 degrees relative to each other. Similarly, all the FGM plates evaluated will
have varying distributions of the epoxy and graphite throughout their thicknesses. For this
project, it is assumed that an unlimited range of varying material distribution can be
achieved using 3D printing to manufacture the FGM plates.
3
2. Methodology
2.1 Methodology Overview
This project was built upon the work that Kenneth Carroll and Kevin Pendley did
in their Master’s Project. In “Comparative Deflection Analysis of Aluminum and
Composite Laminate Plates Using the Rayleigh-Ritz and the Finite Element Method” [6],
Carroll compared the deflection of composite laminate plates with the deflection of a thin
Aluminum plate. The analysis for the composite plate was done using the finite element
method (FEM) in ANSYS as well as using analytical solutions in Maple.
For this project the laminated plates were first analyzed in ANSYS. Then the
solution was validated using Maple to solve the analytical solution. Once the modeling
methodology in ANSYS was validated by using the analytical methods, the same
modeling methodology was used to analyze the FGM plates in ANSYS.
The properties of the FGM plate were approximated using the Mori-Tanaka
method referenced in “Modal Analysis of Simply Supported Functionally Graded Square
Plates” [7]. This methodology uses the material properties and the volumetric ratio of each
element making up the plate to calculate the properties of the functionally graded plate
through its thickness.
2.2 Laminated Ply Properties
To analyze the deflection of CLP plates analytically the elastic modulus, shear
modulus, and Poisson’s ratio of the composite ply were to calculate the Reduced Stiffness
Matrix, which relates the laminate stress to the laminate strain. The Reduced Stiffness
Matrix was used to calculate the CLP deflection via the Classical Lamination Theory for
the cross ply plate [6].
Because the thicknesses of the CLP plates analyzed was much smaller than their
length and width, and the deflection of the plate was small, it was possible to take
advantage of the Thin Plate Theory assumptions to simply the Reduced Stiffness Matrix
from a 6x6 to a 3x3 matrix. These assumptions are that the middle plane of the plate does
not deform but stays neutral after bending, that straight lines that are normal to the middle
4
plane remain straight and normal to the middle plane after bending, and that stresses in the
transverse direction of the plate are low compared to in plane stresses and can therefore
be disregarded [6]. As seen in equation 1 below [6], the 3x3 Reduced Stiffness Matrix
required the values of the elastic modulus in the x and y direction, the Poisson’s ratio in
the xy direction and the shear modulus in the xy direction. Symbolically these are E1, E2,
ν12, and G12, where x has been replaced with the subscript 1 and y with 2. The value for
ν21, which is also needed for the analytical solution is calculated using equation 2 [6].
𝑸𝟏𝟏 =
𝑬𝟏
𝟏−𝒗𝟏𝟐 𝒗𝟐𝟏
𝑸𝟐𝟐 =
𝑸𝟏𝟐 =
𝒗𝟏𝟐 𝑬𝟐
𝟏−𝒗𝟏𝟐 𝒗𝟐𝟏
𝑬𝟐
𝒗𝟐𝟏 𝑬𝟏
𝟏−𝒗𝟏𝟐 𝒗𝟐𝟏
(1)
𝑸𝟔𝟔 = 𝑮𝟏𝟐
𝟏−𝒗𝟏𝟐 𝒗𝟐𝟏
𝒗𝟐𝟏 = 𝒗𝟏𝟐
=
𝑬𝟐
(2)
𝑬𝟏
The properties used for this project come from Appendix C of “Mechanics of
Composite Structures” by L. P. Kollár and G. S. Springer. This text provided values for
E1, E2, ν12, ν23, and G12. However, in order to conduct the FEM analysis in ANSYS the
values of E3, ν13, G23, and G13 were also required. These values were calculated by
assuming transverse isotropy. Transverse isotropy states that for a unidirectional ply the
moduli in the directions perpendicular to the fiber direction are roughly equal and much
smaller than the modulus in the fiber direction. Symbolically that is E1 >> E2 and E3 ~ E2.
Along the same lines ν13 = ν12, and G13 = G12 [8]. Lastly, by using the same assumption,
the value for G23 is calculated via equation 3 [6].
𝑮𝟐𝟑 =
𝑬𝟐
𝟐∗(𝟏+𝒗𝟐𝟑 )
5
(3)
A complete list of the values used for the CLP plate analysis is shown in table 1.
The properties listed in this table are for the AS/3501 (Graphite/Epoxy) combination. The
table also contains the thickness of a laminated ply as well as the fiber volume fraction,
which will also be used in the analysis.
Table 1: Properties of AS/3501 Composite Ply
Property
Value
Comment
E1 (Pa)
138E+9
from text [9]
E2 (Pa)
8.96E+9
from text [9]
E3 (Pa)
8.96E+9
calculated
ν12
0.3
from text [9]
ν23
0.59
from text [9]
ν13
0.02
calculated
G12 (Pa)
7.1E+9
from text [9]
G23 (Pa)
2.82E+9
calculated
G13 (Pa)
7.1E+9
calculated
tply (m)
1.27E-4
from text [9]; ply thickness
Vf (%)
66
from text [9]; fiber volume fraction
2.3 Laminated Plate Modeling
For this project the plates were modeled using SHELL181 element in ANSYS.
This type of element has 4 nodes with six degrees of freedom at each node and provides
the opportunity to add layers. [6]. Therefore once the material properties of a single ply
were entered, the plate was constructed in ANSYS by stacking the correct number layers,
each representing a ply, in the correct angular orientation.
The layers were laid out in a [0 0 90 90 0 0 90 90 0 0]s orientation for the cross
ply CLP plate and in a [0 0 45 45 0 0 -45 -45 0 0]s orientation for the angle ply CLP plate.
The numbers in the brackets represent the order and angle of the ply layer while the letter
“s” at the bracket end implies symmetry. Therefore, the first two layers of the cross ply
6
plate are oriented at 0 degrees, the next two are oriented at 90 degrees from the first two,
and so on. Layers 11 through 20 are symmetric to layers 10 through 1 respectively. Figures
3 and 4 show the ply arrangement for the CLP plates. In this figure the solid blue layers
represent the 0 degree plies while the red pattern represent the 90 degree or ±45 degree
angles plies for the CLP plates 1 and 2 respectively.
Figure 3: CLP Plate Layer (Ply) Arrangement
Figure 4: CLP Plate Layer (Ply) Arrangement
7
For this project all the plates will be square with constant side lengths of a = 0.2
and a height of H = 0.00254 meters, which was derived from having 20 layers with tply =
0.000127 m. These dimensions were chosen such that the plate size is manageable and can
be easily tested in a rig for the purpose of validating the results of the FEA analysis. The
sheet body plate was then meshed using quadrilateral areas and a “smart size” of 3. Figure
4 shows a screenshot of the meshed ANSYS model geometry. Finally loads and boundary
conditions were applied to the sheet body representing the CLP plate to simulate a simply
supported plate with a uniform load applied to it. To achieve this loading condition all
sides were constrained in the z direction. Then sides 1 and 3 were prevented from rotating
about the y axis and sides 2 and 4 were prevented from rotating about the x axis. To fully
constrain the model side 1 was prevented from moving along the y direction and side 4
was prevented from moving in the x direction. Lastly the pressure load of 20 kilo Pascale
is applied evenly on the surface.
Figure 5: Meshed Sheet Body Representing CLP Plate
8
Since entering properties for 20 plies is laborious a simplification was attempted
on the model. A 20 layers model with a layer thickness of 0.000127 m was compared to a
10 layer model with layer thickness of 0.000254m. This simplification was justifiable
because layer orientation changes with every two layers which allowed for the
combination of layer couples with the same orientation into one thicker layer. Simplifying
the model to a ten layer configuration allowed for an easier comparison using Classical
Lamination Theory for a 10 layer cross ply plate. The results of these calibration runs will
be explained in detail in section 3 “Results and Discussion”.
2.4 Properties of FGM Plate Elements
The modulus of elasticity of the graphite reinforced FGM plates is lower relative
to the CLP plate. This reduction in modulus happens because the graphite in the FGM
plate is deposited in grains, thus losing some of the high strength capability derived from
a fibrous condition. Since the textbook used to obtain the laminate properties did not have
properties for graphite grains or chopped fiber, the properties were obtained from Zoltec
Companies online brochure. Before the properties for chopped carbon fibers were used
for the graphite in the FGM plate, the properties for continuous tow carbon fiber and
unidirectional prepreg from Zoltec’s brochures were compared with fiber properties and
laminate ply properties from the textbook to validate that the properties used were
somewhat consistent. The modulus values for unidirectional and continuous fibers and for
unidirectional ply from both Zoltek’s brochure and the textbook matched well. This
closeness in the material tensile modulus properties from the two sources acts as a
validation for using materials from two different sources with the assumption that the
materials whose properties are provided are similar enough. This comparison is captured
in table 2.
Table 2: Young’s Modulus Comparison Between Two Sources
AS4,
9
Panex 35,
% Difference
Text [9]
Zoltek [10]
Continuous Fiber E (Pa)
234E+9
242E+9
2.5
Unidirectional Ply E (Pa)
138E+9
134E+9
2.9
The only value used from Zoltek’s brochure is Young’s modulus for the chopped
fiber. The brochure did not provide a value for Poisson’s ratio for chopped fiber, therefore
the value used for analysis was for unidirectional fibers and came from the textbook, or
source 1. In addition the matrix or epoxy properties also came from the text book. Table
3 summarizes the values from the two different sources and the comparison between the
two.
Table 3: Properties of FGM Plate Substrates
Material
Property
Value
Source
23E+9
Zoltek [10]
Graphite,
Panex 35
Er or E1 (Pa)
Chopped Fiber
AS4
νr or ν1
Epoxy Matrix, 3501
FGM Plate
0.26
text [9]
4.4E+9
text [9]
νm or ν2
0.36
text [9]
tplate (m)
1.27E-4
text [9]
Vf (%)
66
text [9]
Em or E2 (Pa)
2.5 Properties of FGM Plate & Mori-Tanaka Method
The properties of the FGM plate were estimated via the Mori-Tanaka method. This
method calculates the through thickness material properties by using the through thickness
volume fractions of each component making up the FGM plate [7]. It divides the plate into
a chosen number of isentropic layers and calculates the properties for each layer based on
the volume fraction of the constituents for that layer. In the Mori-Tanaka based equations
below the subscript “p” stands for plate, “L” for layer, “1” for material 1 which is the
graphite, and “2” for material 2, which is the epoxy. The two material properties needed
for each layer are Young’s Modulus and Poisson’s ratio. The functions representing these
properties through the FGM plate thickness are:
10
𝑬𝒑 = 𝟑 ∙ 𝑲𝒑 ∙ (𝟏 − 𝟐 ∙ 𝝂𝒑 )
𝟏
𝝂𝒑 =
𝑮𝒑
𝟐∙(𝟏+
𝝀𝒑
(4)
(5)
)
Gp is the equation representing the through thickness shear modulus of the FGM plate and
it is denoted by:
𝑮𝒑 = 𝑮𝟏 +
(𝑮𝟐 −𝑮𝟏 )∙𝑽𝟐
(𝟏+
(𝟏−𝑽𝟐 )∙(𝑮𝟐 −𝑮𝟏 )
)
𝑮𝟏 +𝒇𝟏
(6)
Where ƒ1 is represented by:
𝒇𝟏 =
𝑮𝟏 (𝟗 ∙ 𝑲𝟏 + 𝟖 ∙ 𝑮𝟏 )
𝟔 ∙ (𝑲𝟏 + 𝟐 ∙ 𝑮𝟏 )
λp is Lamѐ first parameter, represented by:
𝟐
𝝀𝒑 = 𝑲 𝒑 − ∙ 𝑮 𝒑
𝟑
(7)
Kp is the bulk modulus of the plate through its thickness and it is represented by the
equation:
𝑲𝒑 = 𝑲𝟏 +
(𝑲𝟐 −𝑲𝟏 )∙𝑽𝟐
(8)
(𝟏−𝑽𝟐 )∙(𝑲𝟐 −𝑲𝟏 )
(𝟏+
)
𝟒
𝑲𝟏 +( )∙𝑮𝟏
𝟑
V2 is the equation representing the volume fraction of material 2 through the plate
thickness and K1 and K2 are the bulk moduli of materials 1 and 2. They are represented
by:
𝑲𝒏 =
𝑬𝒏
𝟑∙(𝟏−𝟐∙𝝂𝒏 )
(9)
G1 is the shear modulus of material 1 and it represented by:
𝑮𝒏 =
𝑬𝒏
𝟐∙(𝟏−𝝂𝒏 )
(10)
The FGM plate is then divided into 10 equally thick layers and the tensile modulus
and Poisson’s ratio for each layer is layer is calculated using equations (4) and (5). For
11
these calculations it is assumed that the tensile modulus and Poisson’s ratio vary linearly
within the layer. This assumption is valid because the layer is very thin with respect to the
plate thickness. Then the values for E and ν were calculated for each layer by solving the
respective equations for the z value representing the middle plane of each layer.
To explore the effects of graphite distribution on plate deflection, several plates
with different through thickness volume fractions were studied. All of these plates
maintained a total volume ratio of 66% reinforcement and 34% matrix and a total plate
thickness of 0.00254 m. These numbers were kept constant to be consistent with the
laminated plates. The first plate that was looked at had an even constant mixture of the
components. Then the volume fraction of matrix (V2) was varied through the plate
thickness via several cosine functions to determine the distribution of graphite that caused
the least amount of deflection. This studied whether placing the graphite on the surface,
where it experiences the highest stress, or on the center, where it experiences the lowest
stress, provided the highest stiffness. The matrix distribution was then also varied linearly
to check whether the graphite provided the highest stiffness when it experienced tensile
stress, meaning when it is on the opposite side the pressure is applied on, or when it
experienced compressive stress. Equation (11) through (15) represents the variation of V2
through the plate thickness via a cosine function.
𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 − 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
)
(11)
)
(12)
𝟐∙𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
𝟑∙𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
𝟒∙𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
)
(13)
)
(14)
)
(15)
Where V20 is a constant that positions the distribution curve above the y=0 axis to make
sure the material ratio is always positive. Va is the amplitude of the cosine function and
the value inside the cosine provides the frequency of the curve. H is the plate thickness
12
which is 0.00254 meters, and “z” is the independent variable that describes the location of
the thickness of the plate. In order to get a plate with constant distribution of the matrix
and reinforcement V20 is set to 0.34 and Va is set to 0.
The equations describing the linear distribution of the matrix through the plate thickness
are described by equation (16) and (17). For the plates where the volume fraction is linear
the value of V20 is both a y intercept value and a contributor to the slope of the line.
𝑽𝟐 = 𝑽𝟐𝟎 ∙ (𝟏 +
𝑽𝟐 = 𝑽𝟐𝟎 ∙ (𝟏 −
𝟐
𝑯
𝟐
𝑯
∙ 𝒛)
(16)
∙ 𝒛)
(17)
The equations describing the sine distribution of the matrix volume through the plate
thickness are described by equations (18) and (19). Similar to the cosine functions, the
sine functions have the values of V20 and Va.
𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐬𝐢𝐧 (
𝑯
𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 − 𝑽𝒂 ∙ 𝐬𝐢𝐧 (
𝑯
)
(18)
)
(19)
Lastly, the volume fraction of the graphite through the plate thickness, or V1, is described
by equation (13).
𝑽𝟏 = 𝟏 − 𝑽𝟐
(20)
Table 4 summarizes the 18 plates analyzed in this study. Column 1 gives the plate label or
number, column 2 describes which equation was used for the volume distribution of the
matrix, and columns 3 and 4 give the values of V20 and Va used. For plates 15 and 16 the
volume distribution of the matrix is linear and therefore Va is not used.
13
Table 4: Summary of Plates Studied
Plate #
V2 Equation
V20
Va
Plate #
V2 Equation
V20
Va
1
(11)
0.000
0.534
10
(13)
0.340
0.050
2
(11)
0.170
0.267
11
(14)
0.430
0.430
3
(11)
0.340
0.000
12
(14)
0.280
0.280
4
(12)
0.500
0.251
13
(15)
0.340
0.340
5
(12)
0.660
0.503
14
(15)
0.340
0.340
6
(12)
0.830
0.770
15
(17)
0.340
N/A
7
(12)
0.935
0.935
16
(16)
0.340
N/A
8
(13)
0.340
0.340
17
(18)
0.340
0.340
9
(13)
0.340
0.160
18
(19)
0.340
0.340
2.6 Functionally Graded Material Plate Modeling
The FGM plates were modeled in ANSYS similar to the CLP plates. The element
chosen was SHELL 181. What differed between the CLP and FGM plates is that the FGM
plate is composed of 10 isentropic layers that differ from each other. For the plates that
have volume fractions represented by the cosine function, the layers are symmetric about
the center plane. For the plates whose volume fractions vary linearly there is no symmetry
between the layers. The layer thickness is kept constant to 0.000254 m, which is equal to
the layer thickness of the laminated plate. This ensures that the total plate thickness
remains constant between the two configuration types. In addition the plate side
dimensions were kept constant at a=0.2 m. The mesh size used was smart size 3 which is
the same mesh size as the CLP plates. Also the same boundary conditions used to constrain
the four sides of the plates and apply the pressure load on the CLP plates were used to
apply the boundary conditions on the FGM plates.
14
3. Results and Discussion
3.1 Laminated Plate Results and Validation of Modeling Method
Two types of laminated plates were studied, a cross ply and an angle ply. In
addition to providing an analysis data point for deflection, the cross ply plate was used to
validate the modeling method. First the cross ply plate was modeled in ANSYS using the
methodology described in section 2.3 of this report with 20 layers as shown by figure 3.
Next, the cross ply plate was modeled assuming that the consecutive plies that were
oriented in the same direction could be combined into one layer with double the thickness,
as is shown by the color coding in figure 3. The plate dimensions, mesh size, and boundary
conditions were not changed. As shown in figures 5 and 6 both the 20 and 10 layer plates
deflect 0.005139 meters. This demonstrates that combining layers with similar
orientations is an acceptable modeling methodology.
Figure 6: Deflection of 10 Layer Cross Ply Laminated Plate
15
Figure 7: Deflection of 20 Layer Cross Ply Laminated Plate
Furthermore the 10 layer plate was analyzed in Maple using the Classical
Lamination theory, as mentioned in section 2.2. The deflection obtained with the Maple
analysis was 0.005149 m which is 0.2% different than the deflection obtained in ANSYS.
This demonstrates the layering, meshing, and boundary application method in ANSYS is
acceptable. Therefore the 10 layer modeling methodology was used to analyze the angle
ply laminate and the FGM plates. The full maple code of the Classical Lamination Theory
is shown in Appendix A.
The angle ply plate with layers at 45 degrees relative to each other was also
analyzed in ANSYS using the 10 layer method. As shown in figure 7, the angle ply plate
deflected 0.004289 m, which is less than the cross ply plate.
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Figure 8: Deflection of 10 Layer Angle Ply Laminated Plate
3.2 Functionally Graded Plate Results
Functionally Graded Plates with eighteen different reinforcement volume
distributions through plate thickness were analyzed to understand what reinforcement
distribution produced the least amount of deflection as well as whether FGM plate
deflections can be comparable to laminated plates. FGM plates 1 through 7 were analyzed
first to understand the effects of reinforcement material distribution on plate deflection.
These 7 plates have matrix material volume distribution represented by equations 11 and
12 which are versions of a half of a cosine curve. Plate 1 is 100% graphite at the surface,
while the rest of the plates employ a gradual shift of the graphite inboard toward the plate
center culminating in plate 7 having almost 0% graphite at the plate surfaces and 100%
graphite in the middle plane. Plate 1 deflects by the least amount and with the gradual shift
17
of graphite to the mid-plane there is also a gradual increase in plate deflection with plate
7 having the highest deflection.
Plates 8 through 14 further explore the effects of graphite distribution through the
plate thickness on plate deflection. The through thickness volume fractions of the epoxy
through these plates are expressed by the cos(nπ) equation where n is 2 for plates 8 through
10, 3 for 11 and 12, and 4 for plates 13 and 14. The conclusion from the analysis of plates
8 through 14 is the same as that of plates 1 through 7. Therefore, for a plate that has
material distribution of elements symmetric about its mid-plane, the least amount of
deflection is experienced when the distribution of reinforcing material, in this case
graphite, is biased towards the plate surfaces.
Lastly plates that have material distribution of elements through the plate thickness
that is not symmetric about the plate mid-plane as represented by equations 16 through
19. These plates were studied to understand whether the stiffness of the plates comes from
having reinforcing material distributed on the surface where the load is applied, on the
opposite side, or both. Biasing the graphite distribution towards the same surface that the
load is applied on produces a lower plate deflection than doing so on the opposite side.
These deflections are in the 0.0167 to 0.0178 m range which is aligned with the symmetric
plates that experienced deflection in the middle of the full range of symmetric plate
deflections.
Table 5 summarizes the FGM plates analyzed, what equations were used to obtain
the volume fraction of matrix (V2) through the thickness, what the fraction of graphite
(V1) is at the plate surfaces and center, and the plate deflection under the applied load.
Figures 8 through 10 show the through thickness volume fractions of the three plates that
deflected the least. Also figures 11 through 13 show the through thickness volume
fractions of the plates that deflected the most. The trend shown by this table and the figures
is that plates the plates with more reinforcing material toward the plate surfaces deflect
less, while plates with more reinforcing material concentrated toward the plate center
deflect more.
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Table 5: Summary of FGM Plate Graphite Distribution and Deflections
#
V2 Equation V1 @ z=+H/2 V1 @ z=0 V1 @ z=-H/2 Deflection (m)
1
(11)
1.000
0.466
1.000
0.013788
2
(11)
0.830
0.563
0.830
0.015257
3
(11)
0.660
0.660
0.660
0.016839
4
(12)
0.500
0.751
0.500
0.018456
5
(12)
0.340
0.843
0.340
0.020196
6
(12)
0.170
0.940
0.170
0.022204
7
(12)
0.065
1.000
0.065
0.023533
8
(13)
1.000
0.320
1.000
0.012647
9
(13)
0.820
0.500
0.820
0.014739
10
(13)
0.710
0.610
0.710
0.016152
11
(14)
0.570
0.140
0.570
0.013366
12
(14)
0.720
1.000
0.720
0.019346
13
(15)
0.320
0.320
0.320
0.018047
14
(15)
1.000
1.000
1.000
0.015223
15
(17)
1.000
0.660
0.320
0.016724
16
(16)
0.320
0.660
1.000
0.017501
17
(18)
0.320
0.660
1.000
0.017785
18
(19)
1.000
0.660
0.320
0.016870
19
Figure 9: Through Thickness Material Fractions of Plate 8
Figure 10: Through Thickness Material Fractions of Plate 11
20
Figure 11: Through Thickness Material Fractions of Plate 1
Figure 12: Through Thickness Material Fractions of Plate 5
21
Figure 13: Through Thickness Material Fractions of Plate 6
Figure 14: Through Thickness Material Fractions of Plate 7
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4. Conclusion
This project analyzed the deflection due to a uniform pressure load of composite
laminated plates and Functionally Graded Material plates. All of the plates analyzed had
the same dimensions and were composed of equal portions of the graphite and epoxy. The
laminated plates were composed of layers of unidirectional graphite fibers and epoxy resin
oriented in [0 0 90 90 0 0 90 90 0 0]s for the cross ply plate and [0 0 45 45 0 0 -45 -45 0
0]s for the angle ply plate. The FGM plates were composed of 3D printed graphite and
epoxy powder. The distribution of the graphite and epoxy through the plate thickness was
represented by a series of cosine, since, and linear functions.
The FGM plates deflected two to five times more than the laminated plates. The
main is that the tensile modulus of the graphite used for the FGM plates was a sixth of the
value used for the laminated ply and a tenth of the tensile modulus of a fiber. The reason
for this debit is that the carbon powder loses a lot of the stiffness achieved by the fiber
form, thus resulting in higher plate deflections.
The deflection of the FGM plates depended heavily on the distribution of graphite
through the plate thickness. The plates that featured higher concentrations of graphite
along the plate surfaces deflected the least amount. These plates had a graphite distribution
that was symmetric about the center plane and spiked out towards the plate surfaces. These
plates deflected on the order of 0.013 to 0.015 m. On the flip side plates that featured
higher concentrations of graphite in the plate center and lower concentrations on the plate
surfaces deflected the most. Plates that had graphite distributions that were symmetric
about the center plane and spiked around the center plane deflected in the range of 0.019
to 0.024 m. There was a third group of plates that deflected in the range of 0.016 to 0.018
m. These were plates with a graphite distribution that was symmetric about the middle
plane and evenly spread as well as plates with an unsymmetrical graphite distribution.
Although the FGM plates experienced higher deflections that the laminated plates,
one has the opportunity to optimize the FGM plates to minimize deflection. This
optimization is possible due to the flexibility in material distribution allowed by the 3D
printing process used to produce the FGM plates. This makes up for the debit incurred
from using non fibrous carbon. As seen by the results of this study when the graphite
distribution is optimized, the best performing FGM plate only deflects three times as much
23
as the best performing laminated plate, even the tensile modulus of its reinforcing material
is only one tenth of the tensile modulus of the reinforcing material in the laminated plate.
24
References
[1]
http://en.wikipedia.org/wiki/Composite_material
Date Accessed: 10/18/2014
[2]
http://en.wikipedia.org/wiki/Composite_laminates
Date Accessed: 10/18/2014
[3]
Faris Tarlochan, “Functionally Graded Material: A New Breed of Engineered
Material”, Journal of Applied Mechanical Engineering, November 2012
http://omicsgroup.org/journals/Functionally-Graded-Material-A-New-Breed-ofEngineered-Material-6832-2168-9873-1-e115.pdf
[4]
http://fugahumana.files.wordpress.com/2012/07/layup1.gif
Date Accessed: 10/18/2014
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6727/015804amj2.jpeg
Date Accessed: 10/18/2014
[6]
Kenneth Carroll, “Comparative Deflection Analysis of Aluminum and Composite
Laminate Plates Using the Rayleigh-Ritz and the Finite Element Method”, RPI
Hartford Master’s Project Fall 2013
http://www.ewp.rpi.edu/hartford/~ernesto/SPR/Carroll-FinalReport.pdf
[7]
Kevin Pendley, “Modal Analysis of Simply Supported Functionally Graded Square
Plates”, RPI Hartford Master’s Project Spring 2014
http://www.ewp.rpi.edu/hartford/~ernesto/SPR/Pendley-FinalReport.pdf
[8]
Mer Arnel Manahan, “A Finite Element Study of the Deflection of Simply Supported
Composite Plates Subject to Uniform Load”, RPI Hartford Master’s Project Fall
2011
http://www.ewp.rpi.edu/hartford/~ernesto/SPR/Manahan-FinalReport.pdf
[9]
László P. Kollár and George S. Springer, “Mechanics of Composite Structures”,
Cambridge University Press 2003, Appendix C
[10] Property brochures for Panex ® 35 Continuous Tow, Chopped Fiber (Type -85), and
Prepreg Tapes from Zoltek Companies website
http://www.zoltek.com/products/panex-35/
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Appendix A
26