Displacement and stress analysis of a functionally graded …cj82ps208/... · DISPLACEMENT AND...
Transcript of Displacement and stress analysis of a functionally graded …cj82ps208/... · DISPLACEMENT AND...
DISPLACEMENT AND STRESS ANALYSIS OF A FUNCTIONALLY GRADED
FIBER-REINFORCED ROTATING DISK WITH NON-UNIFORM THICKNESS AND
ANGULAR VELOCITY
A Thesis Presented
By
Yue Zheng
to
The Department of Mechanical and Industrial Engineering
in partial fulfillment of the requirements
for the degree of
Master of Science
in the field of
Mechanical Engineering
Northeastern University
Boston, Massachusetts
December 2016
ii
ACKNOWLEDGEMENTS
I would first like to express my gratitude to my thesis advisor Prof. H. N. Hashemi for
his patience and continuous support for both my course study and research. The door of
his office is always open, even in his busiest time. I was so inspired by both his
knowledge and life experience that I have kept learning and working.
My sincere thanks also goes to Dr. D. Mousanezhad and Dr. T. Kasikci, who has
helped me a lot by offering pertinent advice on my thesis from their own research
experience.
Last but not the least, I would like to thank my parents Tiwu Zheng and Yonghong
Yue for their support not only for my study but also for my decision to pursue further
knowledge in mechanical engineering at Northeastern. Their love has always been, and
will always be my deepest source of motivation.
iii
ABSTRACT
Displacement and stress fields in functionally graded (FG) fiber-reinforced rotating disks
of non-uniform thickness subjected to angular deceleration are obtained. Two types of
reinforcement are considered: particle-reinforced (PR) and circumferentially-reinforced
(CR). For the former type of disk, the effect of thickness profile, fiber distribution,
deceleration and Von Mises stress are investigated. The effect of fiber distribution,
thermal loading and Tsai-Wu failure criterion are evaluated for the latter. The governing
equations for displacement and stress fields are derived and solved using finite difference
method. A parametric study confirms that a thickness profile of 𝛼
𝑟+ 𝛽 is the optimum in
reducing stress. The results show that, in general, a higher proportion of fiber leads to
lower displacement and stress field due to a stronger material property of fiber, except for
the hoop stress of CR disks. In addition, it is concluded that disk deceleration has no
effect on the radial and hoop stresses. However, it will only affect the shear stress. A
larger deceleration results in a higher shear stress. For the CR disks, the displacement and
stress fields under a thermal loading are obtained. It is found that the radial displacement,
radial stress and hoop stress are substantially higher when temperature rises. Also, Von
Mises stress is evaluated for the PR disks. The location of maximum Von Mises stress
shifts to the inner radius of disk when the deceleration is greater. At last Tsai-Wu failure
criterion is adopted to check the CR disks under various deceleration. In this case, the
deceleration makes littles difference on the Tsai-Wu index due to a far smaller magnitude
in the shear stress compared to the radial and hoop stress. The failure location depends on
the fiber distribution.
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TABLE OF CONTENTS
ABSTRACT ........................................................................................................................ i
1. INTRODUCTION ..................................................................................................... 1
1.1. Literature Review ................................................................................................. 1
1.2. Statement of the Problem ..................................................................................... 3
2. MODELLING AND FORMULATIONS ................................................................ 4
2.1. Governing Equations and Boundary Conditions .................................................. 4
2.2. Particle-reinforced Isotropic Disk ........................................................................ 5
2.3. Circumferentially-reinforced Transversely Isotropic Disk .................................. 8
3. METHOD ................................................................................................................. 12
3.1. Finite Difference Method ................................................................................... 12
3.2. Validation of the Method ................................................................................... 14
4. RESULTS ................................................................................................................. 15
4.1. Particle-reinforced Isotropic Disk ...................................................................... 15
4.1.1. Results for ceramic-rich composites at the inner radius
n
m
r aV
b a
..... 15
4.1.1.1. Effect of Various Thickness Profiles ................................................... 15
4.1.1.2. Effect of the Gradation (Fiber Distribution) ........................................ 21
4.1.1.3. Effect of the Deceleration and Von Mises Stress ................................ 22
4.1.2. Examples for ceramic-rich composites at the outer radius
n
mVb a
b r
. 24
4.2. Circumferentially-reinforced Transversely Isotropic Disk ................................ 26
4.2.1. Effect of the Gradation (without Temperature Change) ............................. 26
4.2.2. Application of Tsai-Wu Failure Criterion ................................................... 30
4.2.3. Effect of Thermal Loading.......................................................................... 34
REFERENCES ................................................................................................................ 37
LIST OF TABLES .......................................................................................................... 40
LIST OF FIGURES ........................................................................................................ 41
1
1. INTRODUCTION
1.1. Literature Review
Functionally graded materials are a new class of composite materials used as an
alternative to traditional materials in engineering devices These materials provide more
resistance to crack initiation and propagation, and have a higher strength-to-weight ratio
compared to their homogenous counterparts [1-4].
There is a number of investigations towards understanding in-plane and out-of-plane
behavior of FG rotating disks made of an isotropic material [5-19]. Çallioğlu et al. [6]
obtained closed-form solutions for stress field in a FG rotating disk with constant angular
velocity. It was found that lower radial displacement is obtained by increasing elastic
modulus from inner radius to the outer radius. In contrast both radial and hoop stresses
increase by increasing elastic modulus from inner radius to the outer radius. Dai et al. [7]
studied stress field in a decelerating FG disk with uniform thickness using Runge-Kutta
method. They observed the existence of shear stress and showed that stresses decrease
when the gradient index increases. Bayat et al. [8, 9] obtained solutions for stress field in a
FG rotating disk with non-uniform thickness for ‘free-free’ and ‘fixed-free’ boundary
conditions. They found that a FG rotating disk with parabolic or hyperbolic convergent
thickness profile has smaller stresses and displacements compared with that of uniform
thickness. Horgan and Chan [19] found that for a inhomogeneous disk, the location of
maximum radial and hoop stresses shift compared to the homogeneous disk problem.
Similar investigations have been carried out by Damircheli et al. [10], Hassani et al. [11],
Afsar et al. [12, 13], and Go et al. [13], to obtain stress field in FG rotating disks under
thermal loads. Also, vibrations of FG disks have been studied by numerous researchers
[14-18]. Asghari and Ghafoori [20] obtained semi-analytical three-dimensional solutions
for thick FG rotating disks by modifying the two-dimensional plane-stress solutions which
had been previously proposed for thin FG rotating disks. Kadkhodayan and Golmakani [21]
conducted a nonlinear bending analysis on a FG rotating disk, and presented a parametric
study on the effects of material gradient index, angular velocity, and disk geometry on
deformation pattern.
2
Fiber-reinforced composites are a class of FG materials that utilize the combined
stiffness of fiber and flexibility of the matrix to achieve higher performance compared to
homogenous materials. The mechanical behavior of anisotropic rotating disks has been the
subject of a number of studies. Tang [22] obtained closed-form solutions for elastic stress
field in rotating disks with uniform thickness and different boundary conditions and gave
numerical examples. Peng and Li [23] presented an alternative method, which transforms
the arbitrarily-variable-gradient problem into a Fredholm integral equation, to investigate
the effect of orthotropic material properties on stress field in FG rotating disks and verified
the method with numerical results. Murthy et al. [24] developed an analytical solution for
annular disks with non-uniform thickness for different boundary conditions and gave
numerical examples of stress behavior. Reddy et al. [25] presented closed-form solutions
for displacement and stress fields in a FG rotating disk with non-uniform thickness and
density. It is found that the stresses and displacement are lower when the mass density
increases radially. Tahani et al. [26] developed a semi-analytical method to analyze
displacement and stress fields in circumferentially fiber-reinforced composite rotating
disks. Alexandrova et al. [27] presented a mathematical model for displacement and stress
fields in elastic-perfectly plastic anisotropic annular disks. Sayer et al. [28] performed
thermo-elastic stress analysis on a thermo-plastic curvilinearly fiber-reinforced composite
disk. They found that disks under linearly changing temperature produce lower stresses
than under uniform temperature.
To the best of my knowledge, there is no previous research on displacement and stress
fields in FG fiber-reinforced tapered disks with variable thickness and angular velocity in
the literature. This paper presents displacement and stress fields of two kinds of such disks:
particle-reinforced (PR) and circumferentially-reinforced (CR). Two cases of volume
fraction for each kind of disk are investigated. Furthermore, Von Mises Stress for the PR
disks and Tsai-Wu failure criterion for the CR disks are evaluated.
3
1.2. Statement of the Problem
Fig. 1 shows a schematic diagram of the geometry of the disk with inner radius, 𝑟𝑖, and
outer radius, 𝑟𝑜. The disk is rotating with a variable angular velocity (i.e., with angular
acceleration) around the z-axis.
Figure 1. Schematic diagram of tapered disk with non-uniform thickness, where a and b are inner and
outer radius respectively. The disk is rotating around z axis.
The volume fraction of the fiber and matrix are related as:
𝑉𝑓 + 𝑉𝑚 = 1 (1)
where 𝑉𝑓 is studied in two cases. The first one is fiber-rich composites at the outer radius,
while the second one is fiber-rich composites at the inner radius. For each case, the fiber
volume fraction at any location is assumed to be a power-law function of the radial
coordinate, r, as the following:
1 , 0
n
f
r a
bnV
a
(2)
2 , 0
n
f
b rnV
b a
(3)
where subscripts 1 and 2 represent case 1 and case 2, and n is the gradient index
representing the level of material gradation. For n = 0, 𝑉𝑓 becomes equal to unity, which
means that the disk is made of fiber material (i.e., 𝑉𝑚 = 0), while for other values of 𝑛, the
disk is made of a fiber-reinforced composite material (i.e., FG material).
The rest of the paper is organized as follows: The material properties are described and
the governing equations are derived in Section 2. The finite difference method (FDM),
used to solve the governing equations, as well as the boundary conditions is introduced and
validated in Section 3, and finally, results and discussions are presented in Section 4.
4
2. MODELLING AND FORMULATIONS
2.1. Governing Equations and Boundary Conditions
Figure 2. An element of the disk with all in-plane tractions presented in a polar coordinate system.
Fig. 2 shows an element of the disk with all in-plane tractions represented in a polar
coordinate system located at the center of the disk, where r and 𝜃 are the radial and
angular coordinates. Assume that the disk is thin, having a variable thickness, h, which is
a function of r [ℎ = ℎ(𝑟)], and it is rotating with a variable angular velocity, 𝜔(𝑡). The
equilibrium equations for the element shown in Fig. 2 is given as the following:
2 2r
rrh r r h r h r r t r h rr
(4)
2
r r
th r r h r r r h r
r t
(5)
where 𝜎𝑟𝑟 and 𝜎𝜃𝜃 are normal stresses in radial and circumferential directions,
while 𝜏𝑟𝜃 is the in-plane shear stress, and time is denoted by t.
The disk is assumed to be fixed to a circular shaft, and free at the outer edge.
Therefore the boundary conditions can be presented as:
0ru a , 0rr b (6)
0u a , 0r b (7)
where 𝑢𝑟 and 𝑢𝜃 are radial and circumferential displacement, and 𝜎𝑟𝑟 and 𝜏𝑟𝜃 are radial
stress and shear stress.
5
2.2. Particle-reinforced Isotropic Disk
In this model, it is assumed that the disk is made of ceramic-particle-reinforced
composite with material properties at any location obtained by the rule of mixture. Based
on this rule, the elastic modulus, E, and mass density, 𝜌, are given as:
𝐸 = 𝐸𝑚𝑉𝑚 + 𝐸𝑐𝑉𝑐 (8)
𝜌 = 𝜌𝑚𝑉𝑚 + 𝜌𝑐𝑉𝑐 (9)
where the subscripts, m and c, stand for metal and ceramic, respectively. 𝑉𝑚 and 𝑉𝑐 are
respectively the volume fraction for metal and ceramic. By using Eqs. 1, 2, 3, 8 and 9, the
material properties of the PR disk can be presented as:
𝐸(𝑟) = (𝐸𝑚 − 𝐸𝑐) (𝑟−𝑎
𝑏−𝑎)
𝑛
+ 𝐸𝑐 (10)
𝜌(𝑟) = (𝜌𝑚 − 𝜌𝑐) (𝑟−𝑎
𝑏−𝑎)
𝑛
+ 𝜌𝑐 (11)
and
𝐸(𝑟) = (𝐸𝑚 − 𝐸𝑐) (𝑏−𝑟
𝑏−𝑎)
𝑛
+ 𝐸𝑐 (12)
𝜌(𝑟) = (𝜌𝑚 − 𝜌𝑐) (𝑏−𝑟
𝑏−𝑎)
𝑛
+ 𝜌𝑐 (13)
where the first and second sets of equations respectively present the material properties of
FG disks for case 1 and 2.
Because this is an isotropic disk, the stress-strain relations can be presented as:
𝜎𝑟𝑟 = 𝐸(𝑟) ∙ (𝑐2𝜀𝑟𝑟 + 𝑐1𝜀𝜃𝜃) (14)
𝜎𝜃𝜃 = 𝐸(𝑟) ∙ (𝑐1𝜀𝑟𝑟 + 𝑐2𝜀𝜃𝜃) (15)
𝜏𝑟𝜃 =1
2(𝑐2 − 𝑐1) ∙ 𝐸(𝑟) ∙ 𝛾𝑟𝜃 (16)
where 𝜀𝑟𝑟 and 𝜀𝜃𝜃 are normal strains in radial and tangential directions, while 𝛾𝑟𝜃 is the
in-plane shear strain. Moreover, 𝑐1 =𝜈
1−𝜈2 and 𝑐2 =
1
1−𝜈2, where 𝜈 is the Poisson’s ratio
of both materials, which is assumed to be constant. Now, the state of strain in the element
(shown in Fig. 2) can be presented as:
𝜀𝑟𝑟(𝑟, 𝑡) =𝜕𝑢𝑟(𝑟,𝑡)
𝜕𝑟 (17)
𝜀𝜃𝜃(𝑟, 𝑡) =𝑢𝑟(𝑟,𝑡)
𝑟+
1
𝑟∙
𝜕𝑢𝜃(𝑟,𝑡)
𝜕𝜃 (18)
6
𝛾𝑟𝜃 =1
𝑟
𝜕𝑢𝑟(𝑟,𝑡)
𝜕𝜃+
𝜕𝑢𝜃(𝑟,𝑡)
𝜕𝑟−
𝑢𝜃(𝑟,𝑡)
𝑟 (19)
Now, substituting equations 16-18 into 14-16 will result in the following:
𝜎𝑟𝑟 = 𝐸(𝑟) ∙ (𝑐2𝜕𝑢𝑟
𝜕𝑟+ 𝑐1
𝑢𝑟
𝑟+ 𝑐1
1
𝑟
𝜕𝑢𝜃
𝜕𝜃) (20)
𝜎𝜃𝜃 = 𝐸(𝑟) ∙ (𝑐1𝜕𝑢𝑟
𝜕𝑟+ 𝑐2
𝑢𝑟
𝑟+ 𝑐2
1
𝑟
𝜕𝑢𝜃
𝜕𝜃) (21)
𝜏𝑟𝜃 =𝑐2−𝑐1
2∙ 𝐸(𝑟) ∙ (
1
𝑟
𝜕𝑢𝑟
𝜕𝜃+
𝜕𝑢𝜃
𝜕𝑟−
𝑢𝜃
𝑟) (22)
For axisymmetric disk geometry and material properties, where displacements are
functions of r only, equations 20-22 can be simplified to:
𝜎𝑟𝑟 = 𝐸(𝑟) ∙ (𝑐2𝜕𝑢𝑟
𝜕𝑟+ 𝑐1
𝑢𝑟
𝑟) (23)
𝜎𝜃𝜃 = 𝐸(𝑟) ∙ (𝑐1𝜕𝑢𝑟
𝜕𝑟+ 𝑐2
𝑢𝑟
𝑟) (24)
𝜏𝑟𝜃 =𝑐2−𝑐1
2∙ 𝐸(𝑟) ∙ (
𝜕𝑢𝜃
𝜕𝑟−
𝑢𝜃
𝑟) (25)
Then, substituting equations 23-25 into the governing equations (i.e., Eqs. 4-5) will result
in:
𝜕2𝑢𝑟
𝜕𝑟2 +𝜕𝑢𝑟
𝜕𝑟(
𝑑𝐸
𝐸𝑑𝑟+
𝑑(𝑟ℎ)
𝑟ℎ𝑑𝑟) + 𝑢𝑟 (
𝑐1𝑑𝐸
𝑐2𝐸𝑑𝑟−
𝑐1
𝑐2
1
𝑟2 +𝑐1
𝑐2
𝑑(𝑟ℎ)
ℎ𝑟2𝑑𝑟−
1
𝑟2) = −𝜌𝜔2𝑟
𝑐2𝐸 (26)
𝜕2𝑢𝜃
𝜕𝑟2 +𝜕𝑢𝜃
𝜕𝑟(
𝑑𝐸
𝐸𝑑𝑟+
𝑑(𝑟ℎ)
ℎ𝑟𝑑𝑟) + 𝑢𝜃 (−
𝑑𝐸
𝐸𝑟𝑑𝑟−
𝑑(𝑟ℎ)
ℎ𝑟2𝑑𝑟) =
2𝜌𝑟
(𝑐2−𝑐1)𝐸
𝑑𝜔
𝑑𝑡 (27)
Next, assuming the angular velocity of the disks to be a natural-exponential function of
time, 𝜔(𝑡) = 𝜔0𝑒−𝜆𝑡, where 𝜔0 and 𝜆 are two constant parameters, it can be shown that
the displacement field can be given as the following:
𝑢𝑟(r, t) = 𝑅𝑟(𝑟) ∙ 𝐾𝑟(𝑡) (28)
𝑢𝜃(r, t) = 𝑅𝜃(𝑟) ∙ 𝐾𝜃(𝑡) (29)
where 𝐾𝑟(𝑡) = 𝑒−2𝜆𝑡, 𝐾𝜃(𝑡) = 𝑒−𝜆𝑡, and 𝑅𝑟(𝑟) and 𝑅𝜃(𝑟) are two unknown functions to
be determined. Then, substituting equations 28-29 into equations 26-27, will transform the
governing equations into the following set of ordinary differential equations [which must
be numerically solved for 𝑅𝑟(𝑟) and 𝑅𝜃(𝑟)]:
𝑅𝑟′′(𝑟) + 𝑔1(𝑟)𝑅𝑟
′ (𝑟) + 𝑔2(𝑟)𝑅𝑟(𝑟) = 𝑔3(𝑟) (30)
𝑅𝜃′′(𝑟) + 𝑔4(𝑟)𝑅𝜃
′ (𝑟) + 𝑔5(𝑟)𝑅𝜃(𝑟) = 𝑔6(𝑟) (31)
where
𝑔1 =𝑑𝐸
𝐸𝑑𝑟+
𝑑(𝑟ℎ)
𝑟ℎ𝑑𝑟 (32a)
7
𝑔2 =𝑐1
𝑐2
𝑑𝐸
𝐸𝑟𝑑𝑟−
𝑐1
𝑐2
1
𝑟2 +𝑐1
𝑐2
𝑑(𝑟ℎ)
ℎ𝑟2𝑑𝑟−
1
𝑟2 (32b)
𝑔3 = −𝜌𝜔0
2𝑟
𝐸𝑐2 (32c)
𝑔4 =𝑑𝐸
𝐸𝑑𝑟+
𝑑(𝑟ℎ)
ℎ𝑟𝑑𝑟 (32d)
𝑔5 = −𝑑𝐸
𝐸𝑟𝑑𝑟−
𝑑(𝑟ℎ)
ℎ𝑟2𝑑𝑟 (32e)
𝑔6 =2𝜌𝑟𝜆𝜔0
𝐸(𝑐1−𝑐2) (32f)
Furthermore, the fixed-free boundary conditions, presented in Eqs. 6-7, will be modified
to the following:
𝑅𝑟(𝑎) = 0 (33a)
𝑐2𝑅𝑟′ (𝑏) +
𝑐1
𝑏𝑅𝑟(𝑏) = 0 (33b)
𝑅𝜃(𝑎) = 0 (34a)
𝑅𝜃′ (𝑏) −
1
𝑏𝑅𝜃(𝑏) = 0 (34b)
The equations 30-34 will be numerically solved using the finite difference
method in Section 3.
8
2.3. Circumferentially-reinforced Transversely Isotropic Disk
Figure 3. Schematic diagram of unidirectional fiber-matrix composite, where the fibers are aligned with
direction 1 (circumferential direction), and direction 2 (radial direction) is perpendicular to direction 1.
In this model, the disk is made of a unidirectional fiber-matrix composite material with
fibers oriented in circumferential direction, Fig. 3. Based on these assumptions, material
properties at any location in circumferential (i.e., direction 1) and radial (i.e., direction 2)
directions are obtained by using the rule of mixture, as the following:
1 f f m mE E V E V (35)
2
f m
f m m f
E EE
E V E V
(36)
12 f f m mV V (37)
21 2 1/f f m mV V E E (38)
12
f m
f m m f
G GG
G V G V
(39)
f f m mV V (40)
1
f f f m m m
f f m m
E V E V
E V E V
(41)
2 f f m mE V E V (42)
9
where 𝐸1, 𝐸2, 𝜈12, and 𝜈21 are directional elastic moduli and Poisson’s ratios of the
composite, and 𝐺12 and 𝜌 are shear modulus and mass density. 𝛼1 and 𝛼2 are the thermal
expansion coefficient for circumferential direction and radial direction respectively.
Moreover, 𝐸𝑓, 𝐺𝑓, 𝜌𝑓, and 𝜈𝑓 are elastic modulus, shear modulus, mass density, and
Poisson’s ratio of the fiber, while 𝐸𝑚, 𝐺𝑚, 𝜌𝑚, and 𝜈𝑚 are the corresponding values for
the matrix.
The stress-strain relations can be presented as
122
2 1
rrrr T
E E
(43)
121
1 2
rr TE E
(44)
12
rr
G
(45)
where T(r) represents for the temperature increment at any location.
Considered the state of strain in the element in Fig. 2, and under non-uniform
temperature, 𝜀𝑟𝑟 and 𝜀𝜃𝜃 can be presented as
( , )( , ) r
rr
u r tr t
r
(46)
( , ) 1 ( , )( , ) ru r t u r tr t
r r
(47)
(48)
For symmetric disk material properties, 𝑢𝑟 and 𝑢𝜃 are only function of 𝑟. Eqs. 46-48 are
then simplified to
( , )( , ) r
rr
u r tr t
r
(49)
( , )( , ) ru r tr t
r (50)
(51)
Substituting Eqs. 49-51 into 43-45 results in
, , ,1 r
r
u r t u r t u r t
r r r
, ,r
u r t u r t
r r
10
2 21 12 1
12 21 12 211 1
r rrr
E u E uT T
r r
(52)
12 2 12 1
12 21 12 211 1
r rE u E uT T
r r
(53)
(54)
Substituting Eqs. 52-54 into the governing equations (i.e., Eqs. 4-5) will result in:
2212 212 21 1 12 21 1 21 1 21 1 1
2 2 2 2
2 2 2 2 2 2 2
2 12 21 1 21 1 12 1
2 2 2 2
1( ) ( )
( ) ( )
r rr
ru u dE E d rh dE E E d rh Eu
r r E dr E r hrdr r E rdr E r E hr dr E r E
dE d rh dE E d rh E TdT T
E dr hrdr r E dr E hrdr E r
2 2 21 1 1 21 1 1
2 2
dT E Td E dT
dr dr E dr E dr
(55)
(56)
Similar to Eqs. 26-27, use separation of variables to transform Eqs. 55-56 into the form of
Eqs. 30-31:
'' '
1 2 3( ) ( ) ( ) ( ) ( ) ( , )r r rR r f r R r f r R r f r t (57)
'' '
4 5 6( ) ( ) ( ) ( ) ( ) ( )t t tR r f r R r f r R r f r (58)
where
(59a)
(59b)
2
12 21 2 12 2 23 2
2 2
21 1 21 1 1 21 1 1 21 1 11
2 2 2 2 2
1 ( )
( )
r dE d hr Td dTf T
E E dr hrdr r dr dr
dE E d hr E E Td E dTT
E dr E hrdr E r E dr E dr
(59c)
(59d)
(59e)
12r
u uG
r r
2
12 12
2 2
12 12 12
+d rh d rhu u dG dG r d
ur r G dr hrdr G rdr hr dr G dt
2 21 1 12
1
2 2
d rhdE E
fE dr E r rhdr r
21 1 21 1 21 1 1
2 2 2 2
2 2 2 2
d rhdE E E Ef
rE dr E r E hr dr E r
12
4
12
+d rhdG
fG dr hrdr
12
5 2
12
d rhdGf
G rdr hr dr
11
(59f)
Now the boundary conditions (Eqs. 6-7) become:
(60a)
(60b)
(61a)
(61b)
(The t term in f3 can be seen as a constant during FD computation process.)
06
12
rf
G
0rR a
'2 21 1
12 21 12 21
01 1
r
r
R bE ER b
b
0R a
'
12
10G R b R b
b
12
3. METHOD
3.1. Finite Difference Method
Finite difference method was utilized to solve the problems. Take the PR model as an
example. The governing equations with the boundary conditions given in Eqs. 30-34. The
radial distance, b-a, was divided into p segments, 𝑝∆𝑟 = 𝑏 − 𝑎, p=4000. A mesh
sensitivity analysis was performed to ensure that the results were independent of mesh
size. No exhaustive effort was made to find optimum mesh size. The finite difference
form of Eq. 29 can be presented as:
1 1
1 2 1 32 2 2
1 2 1
Δ ΔΔ Δ Δm m m
f fR R f R f
r rr r r
(62)
where 𝑅𝑚 = 𝑅𝑟(𝑟𝑚), 𝑟𝑚 = 𝑎 + (𝑚 − 1)Δ𝑟, and m is an integer varying between 1 [𝑅1 =
𝑅𝑟(𝑎)] and p+1 [𝑅𝑝+1 = 𝑅𝑟(𝑏)]. The finite difference form of the boundary conditions
(Eq. 33) can be also presented as:
1 0R (63a)
(63b)
Using above formulations, Eqs. 34-35 can be put into a matrix form as:
[𝐴](𝑝+1)×(𝑝+1) ∙ [𝑅𝑟](𝑝+1)×1 = [𝐵](𝑝+1)×1 (64)
where [A] and [B] are known matrices. Eq. 64 was solved to evaluate [Rr], using
MATLAB. 𝑅𝜃 can be evaluated by following the same process of Eqs. 62-64. The only
difference is to use Eqs. 31 and 34.
The exact same process can be applied to solve the CR model. The results were then
post-processed to evaluate displacement and stress. All parameters used in numerical
calculations are presented in Table 1 and Table 2 (unless otherwise stated).
E2
1-u12u21
Rp+1 - Rp
Dr+
u21E1
1-u12u21
Rp+1
b= 0
13
Table 1. Particle-reinforced disk material properties, geometrical characteristics, gradient index, and
angular velocity used for numerical calculations
Young’s modulus - metal 𝐸𝑚 201.04 GPa
Young’s modulus - ceramic 𝐸𝑐 348.43 GPa
Mass density - metal 𝜌𝑚 8166 kg/m3
Mass density - ceramic 𝜌𝑐 2370 kg/m3
Poisson’s ratio for both materials 𝜈 0.3
Inner radius of disk 𝑎 0.02 m
Outer radius of disk 𝑏 0.1 m
Uniform thickness 𝑡0 0.01 m
Gradient index 𝑛 50
Angular velocity constant [𝜔(𝑡) = 𝜔0𝑒−𝜆𝑡] 𝜔0 100 rad/s
Angular velocity constant [𝜔(𝑡) = 𝜔0𝑒−𝜆𝑡] 𝜆 0.5 /s
Table 2. Disk material properties, geometrical characteristics, gradient index, and angular velocity and
deceleration used for numerical calculations [29, 30]
Elastic modulus of fiber 𝐸𝑓 73.1 GPa
Elastic modulus of matrix 𝐸𝑚 3.45 GPa
Mass density of fiber 𝜌𝑓 2550 kg/m3
Mass density of matrix 𝜌𝑚 2250 kg/m3
Poisson’s ratio of fiber 𝜈𝑓 0.22
Poisson’s ratio of fiber 𝜈𝑚 0.35
Shear modulus of fiber 𝐺𝑓 29.959 GPa
Shear modulus of matrix 𝐺𝑚 12.778 GPa
Inner radius a 0.02 m
Outer radius b 0.1 m
Deceleration factor 𝜆 0.5 /s
Angular velocity before deceleration 𝜔0 100 rad/s
14
3.2. Validation of the Method
The results are presented for the time at which the disk is at the beginning of
deceleration (i.e., at t=0), and therefore subjected to maximum stress field. First the
numerical results were validated by being compared with the analytical closed-form
solution presented in the literature for the case of uniform thickness, constant fiber
volume fraction, and constant angular velocity. The results by FDM were in perfect
agreement with the closed-form solutions presented by Tang [22], as shown in Fig. 4.
Here the radial displacement, radial stress, and hoop stress are normalized as,
1 0/r ru u E , 0/r r , and
0/ , where 1 0.3 0.7 =24.345 GPaf mE E E ,
and 2 2 2 2
0 0 00.3 +0.7 234 kPaf mb b .The validated numerical method
can now be extended to further study of both PR and CR disks.
Figure 4. Comparison between our FDM results and analytical solutions by Tang [22], presented for
radial displacement, and radial and hoop stresses versus normalized radial coordinate.
15
4. RESULTS
4.1. Particle-reinforced Isotropic Disk
4.1.1. Results for ceramic-rich composites at the inner radius
n
m
r aV
b a
4.1.1.1. Effect of Various Thickness Profiles
In order to understand the effect of the disk thickness profile on the stress field, three
different types of thickness profiles were selected: (i) ℎ(𝑟) =𝛼
𝑟+ 𝛽 (i.e., a rational
function of r), (ii) ℎ(𝑟) = 𝛽𝑒−𝛼𝑟(i.e., an exponential function of r), and (iii) ℎ(𝑟) = 𝛼𝑟 +
𝛽 (i.e., a linear function of r), where 𝛼 and 𝛽 are constant parameters. Moreover, the
thickness of the disk at the outer radius is assumed to be a fraction of the thickness of a
disk with uniform thickness (𝑡0):
ℎ(𝑟 = 𝑏) = 𝑞 ∙ 𝑡0 (65)
where q is a real number varying between 0 and 1 (0 < 𝑞 < 1), and 𝑡0 is the thickness of
the disk with uniform thickness. Here I set the value of q to be equal to 1/4, 1/2, and 3/4.
Next, considering each tapered disk to have the same weight compared to that of the disk
with uniform thickness (𝑡0), the following equation must hold:
∫ 2𝜋𝑟ℎ(𝑟)𝜌(𝑟)𝑏
𝑎𝑑𝑟 = ∫ 2𝜋𝑟𝑡0𝜌(𝑟)𝑑𝑟
𝑏
𝑎 (66)
Then, the value of 𝛼 and 𝛽 for each thickness profile can be evaluated by solving
equations 65 and 66. In this section, all the results are presented for the time at which the
disk is at the beginning of deceleration (i.e., at t=0).
16
Figure 5 (a-d). Plots of thickness profile, radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with
thickness profile, ℎ(𝑟) =𝛼
𝑟+ 𝛽, for three different values of the thickness at the outer radius, ℎ(𝑏),
compared to a disk with uniform thickness, 𝑡0, and the same mass.
Figs. 5-7 show the thickness profile, radial stress, circumferential stress, and shear
stress, as functions of the normalized radial coordinate, (r-a)/(b-a), for three different
types of thickness profile, respectively. The results are presented for three different
fractions of the outer radius, q=1/4, q=1/2, q=3/4, and compared with the disk with
uniform thickness. Also, all the disks have the same mass. The results reveal that for
three thickness profiles considered in this investigation, the lowest stress field (i.e., radial,
circumferential, and shear stresses) is achieved by having smaller thickness at the outer
radius (i.e., smaller value for q). Furthermore, these stresses are substantially lower
compared to the disk with a constant thickness (i.e., h(r)=t0).
17
Figure 6 (a-d). Plots of thickness profile, radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with
thickness profile, ℎ(𝑟) = 𝛽𝑒−𝛼𝑟, for three different values of the thickness at the outer radius, ℎ(𝑏),
compared to a disk with uniform thickness, 𝑡0, and the same mass.
18
Figure 7 (a-d). Plots of thickness profile, radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with
thickness profile, ℎ(𝑟) = 𝛼𝑟 + 𝛽, for three different values of the thickness at the outer radius, ℎ(𝑏),
compared to a disk with uniform thickness, 𝑡0, and the same mass.
Next, in order to compare the results for disks with different types of thickness profile,
I rearrange the results presented earlier (i.e., the results presented in Figs. 5-7), and compare
the stress field for different types of thickness profile in Figs. 8-10. The results indicate
that the disk with thickness profile in the form of a rational function of the radial coordinate
(i.e., ℎ(𝑟) =𝛼
𝑟+ 𝛽 ) results in the lowest stress field compared to the other thickness
profiles presented here. Moreover, it can be confirmed that as the thickness of the outer
radius increases (i.e., greater p values), the stress field increases and approaches to that of
the disk with uniform thickness for all types of thickness profile.
19
Figure 8 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with the thickness at the
outer radius, ℎ(𝑏) =𝑡0
4, for three different thickness profiles, compared to a disk with uniform thickness,
𝑡0, and the same mass.
Figure 9 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with the thickness at the
outer radius, ℎ(𝑏) =𝑡0
2, for three different thickness profiles, compared to a disk with uniform thickness,
𝑡0, and the same mass.
Figure 10 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with the thickness at the
outer radius, ℎ(𝑏) =3𝑡0
4, for three different thickness profiles, compared to a disk with uniform thickness,
𝑡0, and the same mass.
A similar analysis was performed for FG disks with the same volume (i.e., instead of
identical mass) and having different types of thickness profile. Again, the results
presented in Figs. 11-13 confirm that the disk with thickness profile in the form of a
rational function of the radial coordinate (i.e., ℎ(𝑟) =𝛼
𝑟+ 𝛽) results in the lowest stress
20
field compared to the other two thickness profiles (similar to the results presented in Figs.
8-10 for disks with the same mass).
Figure 11 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with the thickness at the
outer radius, ℎ(𝑏) =𝑡0
4, for three different thickness profiles, compared to a disk with uniform thickness,
𝑡0, and the same volume.
Figure 12 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with the thickness at the
outer radius, ℎ(𝑏) =𝑡0
2, for three different thickness profiles, compared to a disk with uniform thickness,
𝑡0, and the same volume.
Figure 13 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with the thickness at the
outer radius, ℎ(𝑏) =3𝑡0
4, for three different thickness profiles, compared to a disk with uniform thickness,
𝑡0, and the same volume.
21
4.1.1.2. Effect of the Gradation (Fiber Distribution)
In Figs. 14 and 15 we study the effect of gradient index, n, on the stress field by
comparing the stress field in the disks with thickness profile of ℎ(𝑟) =𝛼
𝑟+ 𝛽 and
thickness at the outer radius of 1
2𝑡0 for five different values of the gradient index, n. The
thickness profile ℎ(𝑟) =𝛼
𝑟+ 𝛽 is chosen here, since it has been verified to have the
optimal thickness profile in terms of the lowest stress field. The thickness profile is
adjusted to have FG disks with the same mass and volume, respectively in Figs. 14 and
15. Note that different values of n result in different mass due to the change in mass
density; however, the difference in mass between n=25 and n=100 in Fig. 14 is as low as
5.5%, thus negligible. The results presented in Figs. 14 and 15 clearly show that for both
cases, stresses are lower for greater values of n (i.e., ceramic at the inner radius and
matrix at the outer radius). This is due to the fact that FG disks with greater gradient
index, n, are mostly constructed from ceramic with higher elastic modulus (compared to
the metal).
Figure 14 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with the thickness at the
outer radius, ℎ(𝑏) =𝑡0
2, and the thickness profile, ℎ(𝑟) =
𝛼
𝑟+ 𝛽, for five different values of gradient index,
n, ranging from 25 to 100, where the disks have the same mass.
22
Figure 15 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the inner radius) with the thickness at the
outer radius, ℎ(𝑏) =𝑡0
2, and the thickness profile, ℎ(𝑟) =
𝛼
𝑟+ 𝛽, for five different values of gradient index,
n, ranging from 0.5 to 50, where the disks have the same volume.
4.1.1.3. Effect of the Deceleration and Von Mises Stress
Finally, in order to study the effects of the gradient index, n, as well as the constant
parameter appearing in the angular acceleration of the disk, , we plot the variations of
Von Mises stress along the radius of FG disks with thickness profile of ℎ(𝑟) =𝛼
𝑟+ 𝛽, for
different values of n and , Fig. 16. The results show that greater values of ,
corresponding to greater disk deceleration, shift the maximum Von Mises stress towards
the inner radius, due to greater shear stress induced at the inner radius region. For smaller
decelerations (i.e., smaller values), the maximum Von Mises stress is located
somewhere between the inner and outer radii. The results also show that the Mises stress
is lower for disks with greater gradient index, which is consistent with the results
presented in Fig. 14-15.
23
Figure 16 (a-d). Von Mises stress versus the normalized radial coordinate, (r-a)/(b-a), for FG disks
(ceramic-rich composites at the inner radius) with the thickness at the outer radius, ℎ(𝑏) =𝑡0
2, and the
thickness profile, ℎ(𝑟) =𝛼
𝑟+ 𝛽, for five different values of 𝜆, ranging from 0.5 to 20, and for different
values of gradient index, n, ranging from 1 to 50.
24
4.1.2. Examples for ceramic-rich composites at the outer radius
n
mVb a
b r
Similar analyses were performed considering matrix-rich composites at the inner
radius (or in other words: ceramic-rich composites at the outer radius). Figs. 17 and 18
plot the stress field along the radial direction of FG disks with different thickness profiles
and thickness fractions at the outer radius. The presented results clearly indicate that the
stress distribution pattern is similar to that of case 1 (i.e., ceramic-rich composites at the
inner radius, 𝑉𝑚 = (𝑟−𝑎
𝑏−𝑎)𝑛); however, all stresses are lower compared to that of case 1,
except at the outer radius. This again can be justified by the fact that ceramic-rich regions
have greater elastic modulus, resulting in a reduced deformation at the outer radius of the
disk and higher hoop stress at the outer radius. Furthermore, the results still indicate that
the optimum stress field can be achieved by selecting the thickness profile in the form of
a rational function of the radial coordinate, ℎ(𝑟) =𝛼
𝑟+ 𝛽.
Figure 17 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the outer radius) with thickness profile,
ℎ(𝑟) =𝛼
𝑟+ 𝛽, for three different values of the thickness at the outer radius, ℎ(𝑏), compared to a disk with
uniform thickness, 𝑡0, and the same mass.
Figure 18 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the normalized radial
coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the outer radius) with the thickness at the
outer radius, ℎ(𝑏) =𝑡0
2, for three different thickness profiles, ℎ(𝑟), compared to a disk with uniform
thickness, 𝑡0, and the same mass.
25
Finally, the von Mises stress is evaluated at different angular decelerations and is
shown in Fig. 19. It can be seen that greater values of 𝜆, corresponding to greater
deceleration, shifts the maximum von Mises stress towards the inner radius due to greater
shear stress (similar to case 1). Also a smaller stress field can be obtained with a greater
value of the gradient index, n. In general, von Mises stress is lower, compared to that of
case 1 (Fig. 16).
Figure 19 (a-d). Von Mises stress versus the normalized radial coordinate, (r-a)/(b-a), for FG disks
(ceramic-rich composites at the outer radius) with the thickness at the outer radius, ℎ(𝑏) =𝑡0
2, and the
thickness profile, ℎ(𝑟) =𝛼
𝑟+ 𝛽, for five different values of 𝜆, ranging from 0.5 to 20, and for different
values of gradient index, n, ranging from 1 to 50.
26
4.2. Circumferentially-reinforced Transversely Isotropic Disk
Since the optimum thickness profile has already been found in Section 4.1, the
thickness h(r) in this CR model is chosen as, 𝛼 𝑟⁄ + 𝛽, where 𝛼 = 0.00075 𝑚2, 𝛽 =
0.0025 𝑚, and 𝑎 ≤ 𝑟 ≤ 𝑏.
4.2.1. Effect of the Gradation (without Temperature Change)
Fig. 20 shows the variations of fiber volume fraction along the radial direction (i.e.,
Eqs. 2 and 3), for different values of gradient index n, confirming the concentration of
fiber density at the outer and inner radii for case 1 and case 2 of FG fiber-reinforced
disks, where the fiber concentration becomes more pronounced for greater values of n.
Figure 20 (a) and (b). Fiber volume fraction distribution along the disk radius for disks with fiber-rich at
the outer and inner radii, presented for different values of gradient index, n, compared to a homogenous
disk with same volume.
Figs. 21-25 show the distribution of radial and circumferential displacements, ru and
u , and radial, circumferential, and shear stresses, rr , , and r , as functions of the
normalized radial coordinate, (r-a)/(b-a) or (b-r)/(b-a), for different values of the gradient
index, 𝑛, for disks with fiber-rich at the outer radius [part (a)], and fiber-rich at the inner
radius [part (b)].
27
Figure 21 (a) and (b). Radial displacement along the disk radius for disks with fiber-rich at the outer and
inner radii, presented for different values of gradient index, n, compared to a homogenous disk with same
volume.
Figure 22 (a) and (b). Circumferential displacement along the disk radius for disks with fiber-rich at the
outer and inner radii, presented for different values of gradient index, n, compared to a homogenous disk
with same volume.
From Figs. 21-22, it can be seen that for both cases of fiber rich at outer and inner
radii, a smaller gradient index n leads to a reduced displacement field due to having over
all more fibers in the disk. The composite with fiber rich at outer radius (Figs. 21a and
22a) exhibits lower displacement field in radial direction but a higher one in
circumferential direction, compared to the disk with fiber rich at inner radius. These
results could be justified considering that the fiber rich zone at the outer radius prevents
radial displacement of the disk. However, fiber rich zone is bonded to the soft matrix
material and disk deceleration results in a tangential force that will simply move this fiber
rich layer in the circumferential direction as a rigid body. This fiber rich layer may not
provide any resistance to the tangential displacement. While in the case of disk with fiber
28
rich at inner radius, the circumferential displacement is significantly lower with the same
values of n, because the inner radius is fixed on the shaft and the fiber rich zone is
substantially more stiff than the matrix, thus resisting circumferential displacement, Fig.
22b. With a smaller value of n, the width of the fiber rich zone increases, enhancing its
resistance to tangential displacement and therefore leads to a lower circumferential
displacement field.
Figure 23 (a) and (b). Radial Stress along the disk radius for disks with fiber-rich at the outer and inner
radii, presented for different values of gradient index, n, compared to a homogenous disk with same
volume.
Figure 24 (a) and (b). Circumferential stress along the disk radius for disks with fiber-rich at the outer and
inner radii, presented for different values of gradient index, n, compared to a homogenous disk with same
volume.
Our results in Fig. 23(a) show that in contrast to a homogenous disk with pure tensile
radial stress at any point, compressive radial stresses can be developed near the outer
radius of the disks with fiber-rich at the outer radius. This compressive radial stress can
29
prevent the propagation of circumferential cracks from initiating at the outer radius of the
disk. In contrast to radial stress, the hoop stress for this type of FG disks is higher
compared to that of homogenous disk, Fig. 24(a).
However, FG disks in this region which has more fibers are much stronger than
homogenous disks. For disks with fiber-rich at the inner radius, the radial stress is lower
in the region close to the inner radius compared to the homogenous disk, Fig. 23(b). The
radial stress distribution then approaches to that of homogeneous disk for all gradient
index. Furthermore, for disks with fiber-rich at the inner radius and gradient index less
than n=5, the hoop stress is lower at the outer radius compared to the homogenous disk.
This will provide a mechanism to mitigate crack initiation and growth in the radial
direction, Fig. 24(b).
For the disk deceleration and geometry used in this study, the shear stress is fairly low
in magnitude and tends to coincides with the shear stress in a homogeneous disk, Fig. 25.
Moreover, the material gradient index, has little effect on the shear stress distribution.
Figure 25. Shear stress along the disk radius for disks with fiber-rich at the outer and inner radii,
presented for different values of gradient index, n, compared to a homogenous disk with same volume.
30
4.2.2. Application of Tsai-Wu Failure Criterion
In order to understand the effect of fiber distribution on the disk failure, Tsai-Wu
failure criterion was adopted in this study [31]. For a disk under plane-stress condition,
Tsai-Wu failure criterion is expressed as the following:
2 2 2
1 2 11 22 66 122rr rr r rrF F F F F F q (67)
where 𝐹𝑖 and 𝐹𝑖𝑗 are strength tensors and are defined as [31]: Composite is assumed
failed when Tsai-Wu index 𝑞 ≥ 1.
1
1 1
t c
FX X
(68)
11
1
t c
FX X
(69)
2
1 1
t c
FY Y
(70)
22
1
t c
FY Y
(71)
66 2
1F
S (72)
12 11 22
1
2F F F (73)
where 𝑋𝑡, 𝑋𝑐, 𝑌𝑡, 𝑌𝑐 are tensile and compressive strength of unidirectional composite in 𝜃
and 𝑟 directions, and S is the shear strength. These strengths strongly depend on fiber
volume fraction. The tensile strength, 𝑋𝑡, in the fiber direction (i.e., circumferential
direction) can be defined as:
mt f m f
f
EX V V
E
(74)
where 𝜎𝑓 is tensile strength of the fiber. The compressive strength in the fiber direction,
𝑋𝑐, assuming transverse fiber buckling mode is given as [32]:
2 13 1
f m fmc f f
f f
V E EEX V V
E V
(75)
31
whereas in the case of fiber buckling in the shear mode, it is given as [32]:
0.63
1
mc
f
GX
V
(76)
where 𝐺𝑚 is shear modulus of the matrix material. In this paper, we assumed fiber
buckling in the shear mode is the dominate mechanism when composite is subjected to
compression in the direction of fibers and used Eq. 46 to evaluate 𝑋𝑐. It has been shown
fiber buckling in the shear mode is the dominant mechanism when fiber volume fraction
is greater than 20% [32].
The composite tensile strength normal to the fiber direction (i.e., radial direction), 𝑌𝑡,
assuming that interfacial strength is stronger than the matrix strength (based on matrix
yielding) is presented as [33]:
11
m yield
t
f m
f m
m
m f f
Y
E E
V
E E V
(77)
where (𝜎𝑚)𝑦𝑖𝑒𝑙𝑑 is the matrix yield strength. Moreover, it can be shown that the
compressive strength normal to the fiber direction, 𝑌𝑐, can be obtained from Eq. 47.
However, we used compressive strength of matrix from Table 2 to evaluate 𝑌𝑐. Finally,
the shear strength of the composite, by assuming that matrix yield first, can be presented
as [32]:
0.5 ( )f
m m fyieldm
GS V V
G (78)
where 𝐺𝑓 is shear modulus of the matrix.
Tsai-Wu failure criterion is evaluated for disks with various fiber volume fraction
distributions (i.e., fiber-rich at the outer and inner radii). Table 3 summarizes the yield
32
strength values used to evaluate Tsai-Wu failure in this study.
Figure 26 (a) and (b). Tsai-Wu Failure Criterion, along the disk radius for disks with fiber-rich at the outer
and inner radii, presented for different values of gradient index, n, compared to a homogenous disk with
same volume.
For the time at which the disk is at the beginning of deceleration (i.e., at t=0), Fig. 26
shows that Tsai-Wu failure criterion is mostly dominated by radial and hoop stresses,
while the shear stress has little effect on the disk failure. Fig. 27 shows the Tsai-Wu
failure criterion for disks with various angular decelerations (i.e., various 𝜆 values). The
results show that Tsai-Wu failure index is weakly dependent on the disk deceleration.
These results can be justified by the fact that both radial and hoop stresses merely depend
on the disk angular velocity. In contrast, the shear stress strongly depends on the disk
angular deceleration, Fig. 28. For all cases, the maximum shear stress is located at the
inner radius, and its value increases with an increase in disk deceleration parameter, 𝜆.
For disks with fiber rich at the inner radius, failure always initiated between inner and
outer radii for any value of gradient index. In contrast, for disks with the fiber rich at the
outer radius, failure location depends on the gradient index. To be specific, for disks with
more reinforcement, 𝑛 ≤ 1, the incipient of the failure is located at the inner region of the
disk. It shifts to the outer radius for 𝑛 > 1. Comparing results for disks with fiber-rich at
the inner radius with those with fiber-rich at the outer radius, it can be concluded that the
latter case with 𝑛 ≤ 1 is more preferable in design.
33
Figure 27. The effect of disk deceleration on the Tsai-Wu failure index for disks with fiber-rich at the outer
radius and gradient index, n=1.
Figure 28 (a-c). Radial, hoop, and shear stress distributions for disks under various angular deceleration.
Table 3. Yield strengths of fiber and matrix used to evaluate Tsai-Wu failure criterion [34, 35].
Tensile strength of fiber (E-glass) 𝜎𝑓𝑡 3445 MPa
Compressive strength of fiber (E-glass) 𝜎𝑓𝑐 1080 MPa
Tensile strength of matrix (Epoxy resin) 𝜎𝑚𝑡 85 MPa
Compressive strength of matrix (Epoxy resin) 𝜎𝑚𝑐 190 MPa
34
4.2.3. Effect of Thermal Loading
Assume the disk is a fin structure since the thickness/radius ratio is far less than 1.
Now consider its inner radius subjected to constant temperature ( ) aT a T , the other
three surfaces subjected to air convection. Then the governing equation of temperature,
T(r), by assuming the disk is a fin structure along the radius since it has a low
thickness/radius ratio, can be derived as:
( )
( ) ( ) 2 ( ) 0d dT r
k r h r r h T r Tdr dr
(79)
where ℎ∞ is the heat convection coefficient, 𝑇∞ is the air temperature and k(r) is the
thermal conductivity along disk radius. The thermal conductivity of the composite,
k(r) is assumed not to vary with respect to the temperature and can be presented as
[36]:
(1 )( )
(1 )
f f m m
m
f m m f
k V k Vk r k
k V k V
(80)
where 𝑘𝑓 and 𝑘𝑚 represents the thermal conductivity of fiber and matrix
respectively.
The temperature T(r) in Eq. 79 can be solved by finite difference method, the
values in Table 4, and by applying one more boundary condition at the outer
surface:
r b
r b
Tk h T T
r
Once the temperature distribution is obtained, the thermal elastic stress can be
achieved by substituting the temperature into Eqs. 52-54. From Eqs. 54 and 56, it
can be seen that the temperature affects neither the circumferential displacement nor
shear stress. As a result, they will remain the same as they were in Fig. 22 and 25.
(Therefore only the radial distribution, radial stress and circumferential stress will be
shown in the following few pages.)
Fig. 29 shows different temperature profiles achieved from different inner
temperature 𝑇𝑎 and same air temperature 𝑇∞ = 25℃.
35
Figure 29 (a) and (b). Temperature profile along the disk radius for disks with fiber-rich at the outer and
inner radii, presented for different values of 𝑇𝑎.
Figs. 30-32 show that the temperature change has a significant effect on the radial
displacement, radial stress and circumferential stress. They are almost 100 times higher
when the temperature rises by 100oC to 200oC, compared to Figs. 21, 23 and 24. When
temperature rises, radial stress becomes negative for most part of the disks, due to the
resistance to expand, from both the fixed side and the outer fiber layer, Fig. 31. Caused
by a similar reason, circumferential stress is negative at inner radius, Fig. 32. Also, since
there is no restriction to the matrix of disks with fiber rich at inner radius, it expands
freely towards the outer radius. So the circumferential stresses, which is subjected to
different temperature profiles, tend to converge to a uniform stress, shown in Fig. 32(b).
Figure 30 (a) and (b). Radial displacement for disks with fiber-rich at the outer and inner radii, presented
for different values of 𝑇𝑎.
36
Figure 31 (a) and (b). Radial stress for disks with fiber-rich at the outer and inner radii, presented for
different values of 𝑇𝑎.
Figure 32 (a) and (b). Circumferential stress for disks with fiber-rich at the outer and inner radii,
presented for different values of 𝑇𝑎.
Table 4. Thermal conductivity, convection coefficient and air temperature used to obtain temperature
profile
Thermal conductivity of fiber (E-glass) 𝑘𝑓 1.3 W/(m oC)
Thermal conductivity of matrix (Epoxy resin) 𝑘𝑚 0.35 W/(m oC)
Heat convection coefficient (Solid - Air) ℎ∞ 5 W/(m2 oC)
Air temperature 𝑇∞ 25 oC
Gradient index n 5
37
REFERENCES
1. Suresh, S. and A. Mortensen, Fundamentals of functionally graded materials,
processing and thermomechanical behavior of graded metals and metal-ceramic
composites. London: IOM Communications Ltd., 1998.
2. Miyamoto, Y., et al., Functionally graded materials: Design, processing and
applications. 1999: Kluwer Academic Publishers, Hingham, MA (US).
3. Mousanezhad, D., et al., Impact resistance and energy absorption of regular and
functionally graded hexagonal honeycombs with cell wall material strain
hardening. International Journal of Mechanical Sciences, 2014. 89: p. 413-22.
4. Ajdari, A., H. Nayeb-Hashemi, and A. Vaziri, Dynamic crushing and energy
absorption of regular, irregular and functionally graded cellular structures.
International Journal of Solids and Structures, 2011. 48(3–4): p. 506-16.
5. Durodola, J.F. and J.E. Adlington, Functionally Graded Material Properties for
Disks and Rotors. Key Engineering Materials, 1997. 127-131: p. 1199-206.
6. Çallioğlu, H., N.B. Bektaş, and M. Sayer, Stress analysis of functionally graded
rotating discs: analytical and numerical solutions. Acta Mechanica Sinica, 2011.
27(6): p. 950-5.
7. Dai, T. and H.-L. Dai, Investigation of mechanical behavior for a rotating FGM
circular disk with a variable angular speed. Journal of Mechanical Science and
Technology, 2015. 29(9): p. 3779-87.
8. Bayat, M., et al., Analysis of functionally graded rotating disks with variable
thickness. Mechanics Research Communications, 2008. 35(5): p. 283-309.
9. Bayat, M., et al., Thermoelastic solution of a functionally graded variable
thickness rotating disk with bending based on the first-order shear deformation
theory. Thin-Walled Structures, 2009. 47(5): p. 568-82.
10. Damircheli, M. and M. Azadi, Temperature and thickness effects on thermal and
mechanical stresses of rotating FG-disks. Journal of Mechanical Science and
Technology, 2011. 25(3): p. 827-36.
11. Hassani, A., et al., Thermo-mechanical analysis of rotating disks with non-
uniform thickness and material properties. International Journal of Pressure
Vessels and Piping, 2012. 98: p. 95-101.
12. Afsar, A.M. and J. Go, Finite element analysis of thermoelastic field in a rotating
FGM circular disk. Applied Mathematical Modelling, 2010. 34(11): p. 3309-20.
13. Go, J., A.M. Afsar, and J.I. Song, Analysis of Thermoelastic Characteristics of a
Rotating FGM Circular Disk by Finite Element Method. Advanced Composite
Materials, 2010. 19(2): p. 197-213.
14. Bahaloo, H., et al., Transverse vibration and stability of a functionally graded
rotating annular disk with a circumferential crack. International Journal of
Mechanical Sciences, 2016. 113: p. 26-35.
15. Pelech, I. and A.H. Shapiro, Flexible Disk Rotating on a Gas Film Next to a Wall.
Journal of Applied Mechanics, 1964. 31(4): p. 577-84.
16. Advani, S.H., Stationary waves in thin spinning disks. International Journal of
Mechanical Sciences, 1967. 9(5): p. 307-13.
38
17. Khorasany, R.M.H. and S.G. Hutton, An analytical study on the effect of rigid
body translational degree of freedom on the vibration characteristics of
elastically constrained rotating disks. International Journal of Mechanical
Sciences, 2010. 52(9): p. 1186-92.
18. Adams, G.G., Critical speeds for a flexible spinning disk. International Journal of
Mechanical Sciences, 1987. 29(8): p. 525-31.
19. Horgan, C.O. and A.M. Chan, The Stress Response of Functionally Graded
Isotropic Linearly Elastic Rotating Disks. Journal of Elasticity, 1999. 55(3): p.
219-30.
20. Asghari, M. and E. Ghafoori, A three-dimensional elasticity solution for
functionally graded rotating disks. Composite Structures, 2010. 92(5): p. 1092-9.
21. Kadkhodayan, M. and M.E. Golmakani, Non-linear bending analysis of shear
deformable functionally graded rotating disk. International Journal of Non-Linear
Mechanics, 2014. 58: p. 41-56.
22. Tang, S., Elastic stresses in rotating anisotropic disks. International Journal of
Mechanical Sciences, 1969. 11(6): p. 509-17.
23. Peng, X.L. and X.F. Li, Elastic analysis of rotating functionally graded polar
orthotropic disks. International Journal of Mechanical Sciences, 2012. 60(1): p.
84-91.
24. Murthy, D.N.S. and A.N. Sherbourne, Elastic stresses in anisotropic disks of
variable thickness. International Journal of Mechanical Sciences, 1970. 12(7): p.
627-40.
25. Reddy, T.Y. and H. Srinath, Elastic stresses in a rotating anisotropic annular disk
of variable thickness and variable density. International Journal of Mechanical
Sciences, 1974. 16(2): p. 85-9.
26. Tahani, M., A. Nosier, and S.M. Zebarjad, Deformation and stress analysis of
circumferentially fiber-reinforced composite disks. International Journal of Solids
and Structures, 2005. 42(9–10): p. 2741-54.
27. Alexandrova, N. and P.M.M. Vila Real, Deformation and Stress Analysis of an
Anisotropic Rotating Annular Disk. International Journal for Computational
Methods in Engineering Science and Mechanics, 2008. 9(1): p. 43-50.
28. Sayer, M., et al., Thermo-Elastic Stress Analysis in a Thermoplastic Composite
Disc. Science and Engineering of Composite Materials, 2005. 12(4): p. 251-60.
29. Sudheer, M., P.K. R., and S. Somayaji, Analytical and Numerical Validation of
Epoxy/Glass Structural Composites for Elastic Models. American Journal of
Materials Science, 2015. 5(3C): p. 162-8.
30. Hosford, W.F., Elementary materials science. 2013: Materials Park, Ohio : ASM
International.
31. Hahn, H.T. and S.W. Tsai, Introduction to Composite Materials. 1980: Taylor &
Francis.
32. Jones, R.M., Mechanics of composite materials. 2nd ed. ed. 1999, Philadelphia,
Pa.: Taylor & Francis.
33. Liu, G., et al., Mechanical Properties of Biomimetic Leaf Composite. ASME's
International Mechanical Engineering Congress & Exposition, 2016(submitted).
39
34. Fiberglass and Glass Technology: Energy-Friendly Compositions and
Applications. 1 ed, ed. P.A.B. Frederick T. Wallenberger. 2010, New York:
Springer US.
35. Simmons: Excellence in Epoxy Resin Worksurfaces. Available from:
http://www.epoxyworktops.com/epoxy-resin/mech-properties.html.
36. Mouritz, A.P. and A.G. Gibson, Fire Properties of Polymer Composite Materials.
1 ed. Solid Mechanics and Its Applications. 2006, Dordrecht: Springer
Netherlands. XII, 401.
40
LIST OF TABLES
Table 1. Particle-reinforced disk material properties, geometrical characteristics, gradient
index, and angular velocity used for numerical calculations ............................................ 13
Table 2. Disk material properties, geometrical characteristics, gradient index, and angular
velocity and deceleration used for numerical calculations [29, 30] .................................. 13
Table 3. Yield strengths of fiber and matrix used to evaluate Tsai-Wu failure criterion
[34, 35]. ............................................................................................................................. 33
Table 4. Thermal conductivity, convection coefficient and air temperature used to obtain
temperature profile ............................................................................................................ 36
41
LIST OF FIGURES
Figure 1. Schematic diagram of tapered disk with non-uniform thickness, where a and b
are inner and outer radius respectively. The disk is rotating around z axis. ....................... 3
Figure 2. An element of the disk with all in-plane tractions presented in a polar
coordinate system................................................................................................................ 4
Figure 3. Schematic diagram of unidirectional fiber-matrix composite, where the fibers
are aligned with direction 1 (circumferential direction), and direction 2 (radial direction)
is perpendicular to direction 1............................................................................................. 8
Figure 4. Comparison between our FDM results and analytical solutions by Tang [22],
presented for radial displacement, and radial and hoop stresses versus normalized radial
coordinate. ......................................................................................................................... 14
Figure 5 (a-d). Plots of thickness profile, radial stress, circumfrential stress, and shear
stress, versus the normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich
composites at the inner radius) with thickness profile, ℎ𝑟 = 𝛼𝑟 + 𝛽, for three different
values of the thickness at the outer radius, ℎ𝑏, compared to a disk with uniform thickness,
𝑡0, and the same mass. ...................................................................................................... 16
Figure 6 (a-d). Plots of thickness profile, radial stress, circumfrential stress, and shear
stress, versus the normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich
composites at the inner radius) with thickness profile, ℎ𝑟 = 𝛽𝑒 − 𝛼𝑟, for three different
values of the thickness at the outer radius, ℎ𝑏, compared to a disk with uniform thickness,
𝑡0, and the same mass. ...................................................................................................... 17
Figure 7 (a-d). Plots of thickness profile, radial stress, circumfrential stress, and shear
stress, versus the normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich
composites at the inner radius) with thickness profile, ℎ𝑟 = 𝛼𝑟 + 𝛽, for three different
values of the thickness at the outer radius, ℎ𝑏, compared to a disk with uniform thickness,
𝑡0, and the same mass. ...................................................................................................... 18
Figure 8 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
inner radius) with the thickness at the outer radius, ℎ𝑏 = 𝑡04, for three different thickness
profiles, compared to a disk with uniform thickness, 𝑡0, and the same mass. ................. 19
42
Figure 9 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
inner radius) with the thickness at the outer radius, ℎ𝑏 = 𝑡02, for three different thickness
profiles, compared to a disk with uniform thickness, 𝑡0, and the same mass. ................. 19
Figure 10 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
inner radius) with the thickness at the outer radius, ℎ𝑏 = 3𝑡04, for three different
thickness profiles, compared to a disk with uniform thickness, 𝑡0, and the same mass... 19
Figure 11 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
inner radius) with the thickness at the outer radius, ℎ𝑏 = 𝑡04, for three different thickness
profiles, compared to a disk with uniform thickness, 𝑡0, and the same volume. ............. 20
Figure 12 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
inner radius) with the thickness at the outer radius, ℎ𝑏 = 𝑡02, for three different thickness
profiles, compared to a disk with uniform thickness, 𝑡0, and the same volume. ............. 20
Figure 13 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
inner radius) with the thickness at the outer radius, ℎ𝑏 = 3𝑡04, for three different
thickness profiles, compared to a disk with uniform thickness, 𝑡0, and the same volume.
........................................................................................................................................... 20
Figure 14 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
inner radius) with the thickness at the outer radius, ℎ𝑏 = 𝑡02, and the thickness profile,
ℎ𝑟 = 𝛼𝑟 + 𝛽, for five different values of gradient index, n, ranging from 25 to 100,
where the disks have the same mass. ................................................................................ 21
Figure 15 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
inner radius) with the thickness at the outer radius, ℎ𝑏 = 𝑡02, and the thickness profile,
ℎ𝑟 = 𝛼𝑟 + 𝛽, for five different values of gradient index, n, ranging from 0.5 to 50, where
the disks have the same volume. ....................................................................................... 22
43
Figure 16 (a-d). Von Mises stress versus the normalized radial coordinate, (r-a)/(b-a), for
FG disks (ceramic-rich composites at the inner radius) with the thickness at the outer
radius, ℎ𝑏 = 𝑡02, and the thickness profile, ℎ𝑟 = 𝛼𝑟 + 𝛽, for five different values of 𝜆,
ranging from 0.5 to 20, and for different values of gradient index, n, ranging from 1 to 50.
........................................................................................................................................... 23
Figure 17 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
outer radius) with thickness profile, ℎ𝑟 = 𝛼𝑟 + 𝛽, for three different values of the
thickness at the outer radius, ℎ𝑏, compared to a disk with uniform thickness, 𝑡0, and the
same mass. ........................................................................................................................ 24
Figure 18 (a-c). Plots of radial stress, circumfrential stress, and shear stress, versus the
normalized radial coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the
outer radius) with the thickness at the outer radius, ℎ𝑏 = 𝑡02, for three different thickness
profiles, ℎ𝑟, compared to a disk with uniform thickness, 𝑡0, and the same mass. ........... 24
Figure 19 (a-d). Von Mises stress versus the normalized radial coordinate, (r-a)/(b-a), for
FG disks (ceramic-rich composites at the outer radius) with the thickness at the outer
radius, ℎ𝑏 = 𝑡02, and the thickness profile, ℎ𝑟 = 𝛼𝑟 + 𝛽, for five different values of 𝜆,
ranging from 0.5 to 20, and for different values of gradient index, n, ranging from 1 to 50.
........................................................................................................................................... 25
Figure 20 (a) and (b). Fiber volume fraction distribution along the disk radius for disks
with fiber-rich at the outer and inner radii, presented for different values of gradient
index, n, compared to a homogenous disk with same volume.......................................... 26
Figure 21 (a) and (b). Radial displacement along the disk radius for disks with fiber-rich
at the outer and inner radii, presented for different values of gradient index, n, compared
to a homogenous disk with same volume. ........................................................................ 27
Figure 22 (a) and (b). Circumferential displacement along the disk radius for disks with
fiber-rich at the outer and inner radii, presented for different values of gradient index, n,
compared to a homogenous disk with same volume. ....................................................... 27
Figure 23 (a) and (b). Radial Stress along the disk radius for disks with fiber-rich at the
outer and inner radii, presented for different values of gradient index, n, compared to a
homogenous disk with same volume. ............................................................................... 28
44
Figure 24 (a) and (b). Circumferential stress along the disk radius for disks with fiber-rich
at the outer and inner radii, presented for different values of gradient index, n, compared
to a homogenous disk with same volume. ........................................................................ 28
Figure 25. Shear stress along the disk radius for disks with fiber-rich at the outer and
inner radii, presented for different values of gradient index, n, compared to a homogenous
disk with same volume...................................................................................................... 29
Figure 26 (a) and (b). Tsai-Wu Failure Criterion, along the disk radius for disks with
fiber-rich at the outer and inner radii, presented for different values of gradient index, n,
compared to a homogenous disk with same volume. ....................................................... 32
Figure 27. The effect of disk deceleration on the Tsai-Wu failure index for disks with
fiber-rich at the outer radius and gradient index, n=1. ...................................................... 33
Figure 28 (a-c). Radial, hoop, and shear stress distributions for disks under various
angular deceleration. ......................................................................................................... 33
Figure 29 (a) and (b). Temperature profile along the disk radius for disks with fiber-rich
at the outer and inner radii, presented for different values of 𝑇𝑎. .................................... 35
Figure 30 (a) and (b). Radial displacement for disks with fiber-rich at the outer and inner
radii, presented for different values of 𝑇𝑎. ....................................................................... 35
Figure 31 (a) and (b). Radial stress for disks with fiber-rich at the outer and inner radii,
presented for different values of 𝑇𝑎.................................................................................. 36
Figure 32 (a) and (b). Circumferential stress for disks with fiber-rich at the outer and
inner radii, presented for different values of 𝑇𝑎. .............................................................. 36