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.

iv

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



thickness profile, ℎ(𝑟) = 𝛽𝑒−𝛼𝑟, for three different values of the thickness at the outer radius, ℎ(𝑏),


18



thickness profile, ℎ(𝑟) = 𝛼𝑟 + 𝛽, for three different values of the thickness at the outer radius, ℎ(𝑏),


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.








outer radius, ℎ(𝑏) =3𝑡0



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).





𝑡0, and the same volume.








outer radius, ℎ(𝑏) =3𝑡0



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).




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




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


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.


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.


coordinate, (r-a)/(b-a), for FG disks (ceramic-rich composites at the outer radius) with the thickness at the


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


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


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


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.

http://www.epoxyworktops.com/epoxy-resin/mech-properties.html

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



composites at the inner radius) with thickness profile, ℎ𝑟 = 𝛽𝑒 − 𝛼𝑟, for three different





composites at the inner radius) with thickness profile, ℎ𝑟 = 𝛼𝑟 + 𝛽, for three different



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




profiles, compared to a disk with uniform thickness, 𝑡0, and the same mass. ................. 19



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




profiles, compared to a disk with uniform thickness, 𝑡0, and the same volume. ............. 20




profiles, compared to a disk with uniform thickness, 𝑡0, and the same volume. ............. 20



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



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



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



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



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

displacement and stress analysis of a functionally graded …cj82ps208/... · displacement and...

Documents