The effect of porosities which can be occurred within FGMs
during the process of sintering has been investigated recently [16, 17,
18, 19]. Due to FGM beams are often used in thermal environment,
investigation on the influence of temperature on free vibration of
FGM beams has been studied by several authors [20, 21]. To the
author’s best knowledge, to the date only Wang and Wu considered
forced vibration of FGM beam in thermal environment under moving
loads [22], in which dynamic response of FGM beam under a
harmonic moving load has been studied by Lagrange method. It
should be emphasized that in the authors [22] studied axially FGM
perfect beams (without porosities), and only the case of uniform
temperature is considered. Mathematically, the uniform temperature
distribution is a special case of nonlinear temperature field, and it is
relatively simple from computational point of view. Investigation on
vibration of FGM beams in thermal environment thus is important
from both the research and practical points of view.

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MINISTRY OF EDUCATION AND
TRAINING
VIETNAM ACADEMY OF
SCIENCE AND TECHNOLOGY
GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY
-----------------------------
BUI VAN TUYEN
DYNAMIC OF FGM BEAMS WITH POROSITIES IN
THERMAL ENVIRONMENT UNDER MOVING LOAD
Major: Engineering mechanics
code: 9520101
SUMMARY OF DOCTORAL THESIS
IN MECHANICAL ENGINEERING AND ENGINEERING
MECHANICS
Hanoi – 2018
The thesis has been completed at: Graduate University Science and
Technology – Vietnam Academy of Science and Technology.
Supervisors: 1. Assoc. Prof. Dr. Nguyen Dinh Kien
2. Dr. Tran Thanh Hai
Reviewer 1: Prof. Dr. Hoang Xuan Luong
Reviewer 2: Assoc. Prof. Dr. Tran Minh Tu
Reviewer 3: Assoc. Prof. Dr. Phan Bui Khoi
Thesis is defended at Graduate University Science and Technology-
Vietnam Academy of Science and Technology ato’clock...’, on
. ,., 2018
Hardcopy of the thesis be found at :
- Library of Graduate University Science and Technology
- Vietnam national library
1
INTRODUCTION
1. Relevance of the thesis topic
The effect of porosities which can be occurred within FGMs
during the process of sintering has been investigated recently [16, 17,
18, 19]. Due to FGM beams are often used in thermal environment,
investigation on the influence of temperature on free vibration of
FGM beams has been studied by several authors [20, 21]. To the
author’s best knowledge, to the date only Wang and Wu considered
forced vibration of FGM beam in thermal environment under moving
loads [22], in which dynamic response of FGM beam under a
harmonic moving load has been studied by Lagrange method. It
should be emphasized that in the authors [22] studied axially FGM
perfect beams (without porosities), and only the case of uniform
temperature is considered. Mathematically, the uniform temperature
distribution is a special case of nonlinear temperature field, and it is
relatively simple from computational point of view. Investigation on
vibration of FGM beams in thermal environment thus is important
from both the research and practical points of view.
2. Thesis objective
This thesis aims to develop finite element models for studying
vibration of FGM porous beams in thermal environment under
moving loads.
3. Scope
The thesis focuses on two-phase transverse FGM beams forming
from ceramic and metal. The external loads considered in the thesis
are the constant speed moving forces and moving harmonic forces.
4. Research methods
Both analytical method and finite element analysis are employed
in the thesis. The analytical method is used to derive equations of
motion for the beam, and the finite element method is then employed
to solve the governing equations and to determine the dynamic
characteristics of the beams.
5. Thesis organization
2
Beside Introduction section, the thesis contains 4 Chapters, a
Conclusion section and a list of publications relevant to the thesis.
References cited in the thesis are listed at the end of the thesis.
CHAPTER 1. OVERVIEW
1.1. FGM beams
Functionally graded materials (FGMs) can be considered as a
new type of composite material. These materials are often formed
from two or more constituent materials whose volume fraction varies
in one or more predefined spatial directions. FGMs overcome the
disadvantage problems such as delamination and stress concentration
which often seen in conventional composite materials. With such
advantages, FGMs have great potential in applications where the
operating conditions are severe, including spacecraft heat shields,
heat exchanger tubes, biomedical implants, flywheels, and plasma
facings for fusion reactors, etc. [24].
FGM beams, the structure considered in this thesis, are assumed
to be formed from two phases, metal and ceramic. The volume
fraction of constituents is considered to vary in a spatial direction,
namely in the beam thickness, by a power-law distribution as [3]
,
1
2
,
2
1
2
n
c c m
h zzV V V
h
h
(1.1)
where Vc, Vm are, respectively, the volume fraction of ceramic and
metal; z is the co-ordinate in the thickness direction, and n is material
grading index, defining the material distribution of the constituents.
In addition to the distribution (1.1), several authors also considered
the variation of the material properties in axial or both axial and
thickness directions.
1.2. Investigations on FGM beams
1.2.1. Mechanical behavior of FGM beams
The traditional analytical methods, especially Galerkin method,
are employed by researchers in studying mechanical behavior of
FGM beams [35-41]. Finite element method (FEM) is also widely
used to study the behaviour of FGM beams. Several finite element
beam models for analysis of FGM beams have been proposed in
3
recent years [59-64], for example the works by Alshorbagy et al. [25]
Mohanty el al. [66, 67], Gan and Nguyen [70, 71, 72]. Eltaher et al.
[73, 74] considered the physical neutral axis position in the
derivation of a finite element model for free vibration analysis of
macro/nano FGM beams. Jin and Wang [76] used the quadrature
method to derive stiffness and mass matrices for free vibration
analysis of FGM beams. Based on the first-order shear deformation
theory, Frikha et al. [77] developed a mixed finite element
formulation for bending analysis of FGM beams.
1.2.2. FGM beams with porosities
Porosities can lower the material stiffness, and as a result they
reduce the ability to resist external loads of structural components.
Wattanasakulpong and Ungbhakorn [18], Wattanasakulpong and
Chaikittiratana [19] proposed a simple model for free vibration
analysis of FGM porous beams, in which the porosity volume is
equally divided to both the ceramic and metal phases. The model has
been employed by Ebrahimi and Zia [79] to study nonlinear free
vibration of FGM Timoshenko beams. Chen et al. [16] proposed a
concept “porosity coefficient” in their study of bending and stability
of FGM porous beams. The model in [16] is then extended by the
authors to nonlinear vibration of sandwich beams with FGM porous
core [80], and free and forced vibration of FGM Timoshenko beams
with porosities [81]. Shafiei and Kazemi [82] studied stability of
nano/micro FGM porous beams with modification of the poroposity
model in [18, 19] by considering non-uniform distribution of
porosities in the beam cross sections. The non-uniform distribution
of porosity model has also been employed to study vibration of 2D-
FGM beams [83].
1.2.3. FGM beams in thermal environment
Chakraborty et al. [84] derived a finite element Timoshenko
beam model for studying wave propagation in sandwich beams with
FGM core in consideration of a uniform temperature rise. Based on
the finite element method, Bhangale and Ganesan [85] investigated
the effect of temperature on natural frequency and loss factor of
FGM sandwich beams with visco-elastic core. Ching and Yen [86]
presented a numerical solution for the themo-mechanical
4
deformation problem of FGM beams. The differential quadrature
method (DQM) has been employed by Xiang and Yang [87] in
studying vibration of non-uniform layer FGM Timoshenko beams
prestress by temperature. Pradhan and Murmu [88] studied free
vibration of FGM sandwich beams resting on an elastic foundation.
DQM was also employed by Malekzadeh [89], Malekzadeh et al.
[90] in free vibration analysis of FGM circular arches and FGM
curved beams in thermal environment. Esfahani et al [92] examined
the influence of elastic foundation support and temperature rise on
the nonlinear stability of FGM Timoshenko beams by the general
DQM. Mahi et al. [30] presented an analytical method to evaluate the
effect of temperature rise on natural frequencies of shear deformable
FGM beams. Wattanasakulpong et al. [21] constructed the governing
equations of thermo-mechanical stability and free vibration of FGM
beams. Ma and Lee [95] proposed an analytical solution for
nonlinear behavior of FGM beam under thermal loading. Analytical
method has also been employed by Eroglu in free vibration analysis
of FGM beams in thermal environment [96]. Trinh et al. [98]
presented an analytical method for vibration and stability analysis of
FGM beams under thermo-mechanical loads. With the aid of Runge-
Kutta method, Kiani et al. [99] examined the effect of environmental
temperature on low velocity impact behavior of FGM beams.
Ghiasian t al. [100] studied static and dynamic stability of FGM
Euler-Bernoulli beams subjected to uniform temperature rise.
Ebrahimi et al. [17] derived equations of motion for studying free
vibration analysis of FGM Euler-Bernoulli beams with porosities in
thermal environment.
1.2.4. FGM beams under moving loads
Lagrange multiplayer method has been employed by Şimşek and
his co-workers in studying vibration of FGM beams excited by
moving loads [4, 5, 6, 8, 10, 11]. Yang et al. [104] studied vibration
of cracked beams under a moving load by assuming an exponential
variation of material properties in the thickness direction. The Ritz
and differential quadrature methods was used by Khalili et al. [105]
to investigate dynamic behavior of FGM beams subjected to a
moving mass. Rajabi et al. [7] employed Petrov–Galerkin method to
transfer a system of the fourth order differential equations of BFGM
5
beams under a moving oscillator to a system of second order
differential equations, and then solving the system by Runge-Kutta
method. Wang and Wu [22] employed Lagrange method to examine
the effect of uniform temperature rise on dynamic bahaviour of
Timoshenko beams formed from axially FGM. Taking the effect of
neutral axial position into account, Gan and Nguyen [106]
formulated a finite beam element for dynamic analysis of multi-span
FGM Timoshenko beams. FEM has also been employed by Gan and
his co-worker in studying axially FGM beams under moving loads
[26], and FGM beams with an immediate support transverse by
moving force [107].
1.3. Studies on FGM beams in Vietnam
Using an analytical m method, Nguyen et al. [111] studied the
static bending and vibration of axially loaded FGM Timoshenko
beam. Bending and vibration of FGM beams have also been
considered by Thai and Vo [112] by different higher-order beam
theories. Based on the third-order shear deformation theory, , Vo et
al. [113] derived equations of motion for FGM sandwich beams with
a homogeneous core, and then employed the finite element method
to compute the natural frequencies of the beams. Vo et al [34]
developed a finite element model for bending and free vibration
analysis of FGM beams. The vibration and diagnosis problems of
cracked beams have been considered by Huyen [114], Huyen and
Khiem [115], Khiem et al [116, 117] by using analytical methods.
Based on the co-rotational approach, Nguyen et al. [118, 119, 120]
derived the finite element models for large displacement analysis of
tapered FGM beams. The finite element method was also employed
by Nguyen and his co-workers to study the large deformation
analysis of FGM frames [121], FGM sandwich frames [33].
Recently, the effect of plastic deformation on stability behavior and
nonlinear bending of FGM beams has been studied by the finite
element method [122, 123, 124].
Vibration of FGM beams under moving loads has been
investigated by several authors in recent years. Trung [13] studied
vibration of FGM beams due to a moving mass or a moving
harmonic load by the finite element method. Ha et al. derived a new
finite element model for vibration analysis of multi-span FGM
6
beams under a harmonic load [14], non-uniform FGM beams under
multiple moving loads [15]. Nguyen et al. [133] used polynomials
derived by Kosmatka to derive a finite element formulation for
vibration analysis of non-uniform FGM beams subjected to a
variable speed moving load. The Kosmatka polynomials have also
been employed by Nguyen et al. [9] in derivation of stiffness and
mass matrices for analyzing 2-D FGM beams under a moving load.
1.4. Comments and study orientation
As can be seen from the above literature review, the
investigation on vibration of FGM beams under moving loads has
been considered by very few authors in recent years. In [12], Ha has
successfully derived finite element formulations for vibration
analysis of FGM beams due to moving loads, but the effect of
porosities and temperature has not been examined. The influence of
porosities and temperature has been considered by several authors,
but only on the free vibration problem. Rigidities and mass moments
of FGM beams are altered when considering the effect of porosities.
In addition, the beams are not only under thermal loading due to the
temperature rise, but the elastic moduli are decreased also. These
factors significantly influence dynamic behavior of the beams, and it
is necessary to take them into consideration. From the above reason,
this thesis investigates the vibration of FGM beams with porosities in
thermal environment under moving loads.
CHAPTER 2. FGM BEAMS IN THERMAL ENVIRONMENT
2.1. FGM beam under moving loads
Fig. 2.1 illustrates a FGM beam with length of L, rectangular
cross section with constant width b and height h. The beam is under
actions of nF loads, F1, F2, FnF, moving from left to right with a
constant speed v. The beam is assumed to be formed from two
constituent materials, ceramic and metal, whose volume fraction
varies according to a power law as
1 , 1
2
n
c c m
zV V V
h
(2.1)
in which Vc , Vm are the volume fraction of ceramic and metal,
respectively; z is a co-ordinate along the thickness direction, and n
7
(nonnegative) is a material index, defining the distribution of the
constituent materials.
x
yz
bL
h
F1F2F nF
y
lç rçng
z,w
MÆt c¾t ngang dÇm
gèm (Ec, Gc, c)
kim lo¹i (Em, Gm, m) b
h
Fig.2.1. FGM beam with porosities under moving loads
2.2. Porosities of FGM beam
In the model in [18, 19], the porosity volume fraction V
(V<<1) is assumed equally divided into ceramic and metal phases.
When the beam is in thermal environment, the material properties of
the FGM beam are evaluated according to
1( , ) ( ) ( )
2
( ) ( ) ( ) 2
c m
m m
n
c
zP z T P T P T
h
VP T P T P T
(2.3)
where Pc and Pm are, respectively, the property of ceramic and metal,
and they depend upon temperature T (K) of the environment; Vα is
the porosity volume fraction.
2.3. Temperature in FGM beam
Temperature distribution along the thickness direction of FGM beam
can be obtained by solving Fourier equation [91, 103]
( ) 0d dTz
dz dz
(2.4)
with boundary conditions T = Tc at z = h/2 and T = Tm at z = - h/2. In
Eq. (2.4), the thermal conductivity κ(z) is assumed to be temperature-
independent. Solving Eq. (2.4) gives the temperature distribution
along the beam thickness in the form
8
/2
/2 /2
( ) /
( ) ( )
z h
m c m
h h
dz dzT T T T
z z
(2.6)
It can be seen from above equation that if Tc = Tm then T = Tc = Tm. In
this case, the temperature is the same at every points of the beam,
and it is called the uniform temperature rise (UTR). In case Tc ≠ Tm,
the temperature is a nonlinear function of the z co-ordinate. The
temperature field thus is a nonlinear temperature rise (NLTR). In this
thesis, a temperature rise T for the NLTR is defined in accordance
with the works in [17, 21], that is T = Tc – Tm = Tc - T0 , with T0 =
300K is the referenced temperature.
2.4. Effect of temperature on material properties
Touloukian [130] shows that a property P of a material depends
on temperature by a nonlinear relation as
1 2 30 1 1 2 3( 1 )P P P T PT PT PT
(2.18)
in which P0, P-1, P1, P2 and P3 the temperature-dependent
coefficients.
Figs. 2.2 and 2.3 illustrate the influence of the porosity volume
fraction Vα and the temperature rise ΔT on Young’s modulus of a
FGM beam formed from stainless steel SUS304 and alumina for
various values of V and for ΔT = 500K. As can be seen from the
figures, Young’s modulus decreases clearly when the effect of
porosities is taken into account, for both the case of UTR and NLTR.
By comparing Fig. 2.2(b) Fig. 2.3, one can see that the effective
Young’s modulus E of the FGM beam decreases more significantly
for the case of UTR.
2.5. Governing equations
2.5.1. Displacement field
Axial and transverse displacements at any arbitrary point of the
beam are given by
0
0
( , , ) ( , ) ( , )
( , , ) ( , )
u x z t u x t z x t
w x z t w x t
(2.24)
9
in which u0(x,t) and w0(x,t) are, respectively, the axial and
transverse displacements of a point on the mid-plane; θ(x,t) is the
cross-sectional rotation, and t is the time variable.
2.5.2. Strain and stress fields
The normal and shear strains resulted from Eq. (2.24) are as
follows
, 0, ,
z , , 0,
xx x x x
x z x x
u u z
u w w
(2.25)
The notation (..),x in the above equation is used to indicate the
derivative with respect to x and (..),z is the derivative with respect to z
variable.
Based on Hook’s law, the normal and shear stresses corresponding to
the strains in (2.25) are given by
0, ,
0,
( , ) ( , ) ( , )
( , ) ( , ) ( , )
xx xx x x
xz xz x
z T E z T E z T u z
z T G z T G z T w
(2.26)
in which E(z,T) and G(z,T) are, respectively, the effective Young’s
modulus and shear modulus, and ψ is the shear correction factor.
-0.5 -0.25 0 0.25 0.5
180
200
220
240
260
280
300
320
340
z/h
E
(G
Pa
)
-0.5 -0.25 0 0.25 0.5
180
200
220
240
260
280
300
320
340
z/h
E
(G
Pa
)
n=5
n=10
n=0.1
n=0.5
n=10
n=1
n=5
n=1
n=0.5
n=0.1
(a) T=0 K,V=0 (b) T=0 K,V=0.1
Fig.2.2. Influence of porosity volume fraction on the effective
Young’s modulus of FGM porous beam
10
-0.5 -0.25 0 0.25 0.5
140
160
180
200
220
240
260
280
z/h
E
(G
Pa
)
-0.5 -0.25 0 0.25 0.5
140
160
180
200
220
240
260
280
z/h
E(
G
Pa
)
n=0.5
n=1
n=5
n=10
n=0.1
n=0.5
n=5
n=10
(a) NLTR, T=500 K, V=0.1 (b) UTR, T=500 K, V=0.1
n=0.1
n=1
Fig.2.3. Influence of temperature on the effective Young’s
modulus of FGM porous beam for UTR and NLTR
2.5.3. Strain energy
The strain energy (U ) can be written in the form
22 211 0, 12 0, , 22 , 33 0,
0
1 2
2
L
x x x x xU A u A u A A w dx (2.27)
in which V in the beam volume, A is the cross-sectional area; A11,
A12, A22 and A33 are, respectively, the axial, axial-bending coupling,
bending and shear rigidities of the beam.
2.5.4. Strain energy due to initial thermal stress
Assuming the beam is free stress at the reference temperature T0
and it is subjected to thermal stress due to the temperature change.
The initial thermal stress resulted from a temperature T is given by
[30, 91]
( , ) ( , )Txx E z T z T T (2.29)
The strain energy caused by the initial thermal stress has the
form [17, 30]
2 20, 0,
0
1 1( , ) ( , )
2 2
L
T x T x
V
U E z T z T Tw dV N w dx (2.30)
where NT is the axial force resultant due to the initial thermal stress.
11
2.5.5. Kinetic energy
The kinetic energy of the FGM with porosities has the form
2 2 211 0 12 0 22
0
0
1 ( ) 2
2
L
I u I u I dxw (2.32)
in which I11, I12 and I22 are the mass moments.
2.5.6. Potential of external loads
The external loads considered in the present thesis a