International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
INVESTIGATIONS ON TRANSIENT
DYNAMIC RESPONSE OF FUNCTIONALLY
GRADED MATERIALS
Palipi Sreenivas1, Dr.A.Siva Kumar2, Subramanyam Pavuluri3, Dr.A.Aruna Kumari4
1M.Tech Student, Department of Mechanical Engineering, MLR Institute of Technology, Dundigal,
Hyderabad-43, Andhra Pradesh, India.
2Professor & HOD, Department of Mechanical Engineering. MLR Institute of Technology, Dundigal,
Hyderabad-43, Andhra Pradesh, India.
3Asst. Professor, Department of Mechanical Engineering. MLR Institute of Technology, Dundigal,
Hyderabad-43, Andhra Pradesh, India.
4Assoc.Professor, Department of Mechanical Engineering. JNTUHCEH, Kukatpally,
Hyderabad-500085 Andhra Pradesh, India.
Abstract
The dynamic response analyses of functionally graded materials subjected to mechanical loads are investigated under ambient
temperature and also deals with the linear vibration of functionally graded material plates in thermal environments. Transient
response analysis is the most general method for computing forced dynamic response. The purpose of transient analysis is to
compute the behavior of a structure subjected to time-varying excitation. The transient excitation is explicitly defined in the time
domain. All of the forces applied to the structure are known at each instant of time. The study of dynamic behavior of FGM plate
is very much important for the design of advanced structure. Finite element method is used to study the static and dynamic
analyses. The governing dynamic equilibrium equations are derived by the Hamilton’s principle. A clamped FGM square plate is
used for the study of direct transient response analysis in thermal environment. In the present analysis, zero initial conditions are
assumed and damping is neglected. The time derivatives in the semi-discrete model were approximated by using the Newmark
direct integration method.
Keywords: Functionally graded plate, Dynamic, Transient response, Newmark integration method.
1. INTRODUCTION
Functionally graded materials (FGMs) are a special kind of composite in which the material properties vary smoothly and
continuously from one material to another. Their applications are especially in high- temperature environments such as
nuclear reactors, chemical plants, rocket heat shields, heat exchanger tubes, heat engine components, skin of supersonic
and hypersonic aerospace vehicles etc. FGMs are microscopically inhomogeneous composites usually made from a
mixture of metals and ceramics. The purpose of the paper is to study the structural behavior subjected to excitation which
varies with time. Comparison studies are provided to verify the stability and accuracy of the method in analyzing the
isotropic plate and two types of FGP's Alumina-Zirconia and Aluminum-Alumina. The influences of the boundary
conditions and volume fraction exponents are also examined in detail.
Japanese material scientist in Yamanouchi et al. [1] has marked the beginning of exploring the possibility of using FGMs
for various structural applications. This grading helps to eliminate the interface problems and mitigating thermal stress
concentrations.
Reddy [2] presented a theoretical formulation and finite element models for FGPs based on the third-order shear
deformation theory.
Zhang and Zhou [3] give theoretical analysis of FGM thin plates based on physical neutral surface. Under the assumption
of poison’s ratio is constant across the thickness and the displacements in surface can be neglected in small deflection
problems, there is no stretching-bending coupling effect in constitutive equations of physical neutral surface theory.
Praveen and Reddy [4] investigated the static and dynamic thermo-elastic response of functionally graded plates (FGPs)
using the finite element method.
Wu Lanhe et al. [5] studied the dynamic stability analysis of FGM plates by using least squares differential quadrature
method.
Huang and Shen [6] studied the nonlinear vibration and dynamic response of functionally graded plates in thermal
environments. Heat conduction and temperature-dependent material properties are both taken into account. The
numerical illustrations concern nonlinear vibration characteristics of functional graded plates with two constituent
materials in thermal environments. The results reveal that the temperature field and volume fraction distribution have
significant effect on the nonlinear vibration and dynamic response of functionally graded plate.
Volume 2, Issue 9, September 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
2. MATHEMATICAL FORMULATION
The governing equation for direct transient analysis without exposed to thermal environment can be obtained Equation,
can be written as
..
(1)
The governing equation for direct transient analysis with exposed to thermal environment can be can be written as
[M ]{d }  [ K ]{d }  {F}
..
(2)
[ M ]{d }  [ K * ]{d }  {F }
The above equations become transient when the mechanical force, {F } varies with time. It is solved by using Newmark’s
method as follows.
Newmark’s Time Integration Scheme
The governing dynamic equation, which is transient in nature. Newmark’s direct integration scheme is employed to
approximate the time derivatives and thereby to solve the forced vibration equations. The method is briefly introduced as
follows, as outlined by Bathe [8]In this method function and its derivative are approximated according to following
assumption
.
.
..
..
(3)
{(U )t t }  {U t }  [ a3{U t }  a4 {(U )t t }]
.
..
..
1
{U t t }  {U t }  {U t }t  [(   ){U t }   {(U )t t }]t 2
2
t = time step, t = step number
For  = 0.5 and  = 0.25, Newmark method is unconditionally stable.
(4)
Primary aim is to calculate displacements {U t t } , the governing equation is evaluated at time tt t as
..
(5)
[M ]{(U )t t } [ K ]{U t t }  {Ft t }
Solution for displacements at time tt t is obtained by rearranging the Equation, such that
..
.
..
(6)
{(U )t t }  a0 [{U t t }  {U t }]  a1{U t }  a2{U t }
where,
a0 
1
1
1
, a1 
, a2 
 1, a3  (1   )t , a4  t
 (t 2 )
 ( t )
2
Equation (50) now substitute back in Eq. (48)
[ K1 ]{U t t }  {F1}t t
(7)
where,
(8)
[ K1 ]  [ K ]  a0 [ M ]
.
.
{( F1 )t t }  {Ft t }  [ M ]( a0 {U t }  a1{U t }  a2 {U t })
(9)
.
Once the displacements {U t t } at time tt t are obtained by solving eq. (7) the velocities (U )t t and accelerations
..
(U )t t are computed respectively.
3. NUMERICAL APPROACH
In this present evaluation, 9-noded quadrilateral C0 isoparametric element with 5 degrees of freedom per node is
employed. The degrees of freedom per node are, {u0 v0 w0 θx and θy }. In the dynamic analysis, zero initial conditions are
assumed and damping is neglected. Reduced Gauss integration is utilized to compute the stiffness matrix to avoid the
shear locking. 3x3 Gauss integration is used to compute the bending stiffness and 2x2 gauss integration is used to
compute the shear stiffness.
For the sake of brevity, a clockwise notation starting from x=0 is employed, symbol "CFCS", for example, identifies a
plate clamped at x = 0, 1, free at y = 1, and simply supported at y = 0. A program was developed, it was coded in
MATLAB and many examples were solved numerically.
Material properties of FGM components , Stainless steel, silicon nitride, zirconia, Al, Alumna, Ti-6Al-4V are chosen at
temperature 300K. Temperature dependent coefficients for ceramics and metals for use in eq. 5, for the materials ZrO2,
Ti-6Al-4V, Si3N4, SUS304 are considered at 300K.
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Volume 2, Issue 9, September 2013
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Numerical results for direct transient response:
Parametric studies have been performed to study the transient response of two types of square FGM plates Al/Al2O3 and
Al/ZrO2 subjected to a suddenly applied uniform pressure loading (qo) -1.0e6 N/m2 . The thickness and side of the square
plate are h= 0.01m and a= 0.2m, respectively. In the present analysis, zero initial conditions are assumed and damping is
neglected. The time derivatives in the semi-discrete model were approximated by using the Newmark direct integration
method. For all the cases, a time step of 0.0001s was used. It should be appreciated that in all these figures,
W  wEm h / q0 a 2 and T  t Em /  m a 2 represent the dimensionless forms of, respectively, time and deflection at
the point (x,y) = (a/2, b/2).
Figures. 1 and 2 shows the direct transient response of cantilever FGM plates- Al/Al2O3 and Al/ZrO2 respectively. As the
volume fraction exponent increases, reduces the bending rigidity of the plate. So, the lower the bending rigidity, the
higher the magnitude of deflection. This is what we observed from the Figs. 1 and 2. At any given n value, the magnitude
of deflection is more in the case of Al/ZrO2 plate when compared to Al/Al2O3, except at n = ∞. The above variation is due
to, Young's modulus of ZrO2is being less when compared to Al2O3. At n = ∞, the magnitude of deflection is same in both
the cases, because the same metal (Al) is used.
Figure 1 Temporal evolution of center deflection
of cantilever FGM plate under suddenly applied
uniform loading of -1.0e6 N/m2 (Al/Al2O3)
Figure 2 Temporal evolution of center deflection
of cantilever FGM plate under suddenly applied
uniform loading of -1.0e6 N/m2 (Al/ZrO2)
Figure 3 Temporal evolution of center deflection of cantilever FGM plate under
suddenly applied uniform loading of -1.0e6 N/m2 (Al/ZrO2)
Transient response of silicon nitride and stainless steel FGM square plate, referred to as Si3N4/SUS304 with clamped
boundary condition is investigated under suddenly applied uniform lateral pressure loading in thermal environment:
The material properties, such as Young's modulus E, the poison's ratio ν and coefficient of thermal expansion α can be
expressed as a non-linear function of temperature, Ref.[14], as
2
3
P  P0 ( P1T 1  1  PT
)
1  P2T  PT
3
in which T=T0+ΔT and T0 = 300 Kelvin (room temperature), where Po, P-1,P1, P2, and P3 are the coefficients of temperature
T (K) and are unique to the constituent materials. Typical values for silicon nitride and stainless steel listed in Table below
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except the poison's ratio ν and whereas the mass density ρ is not a function of temperature. Po , P-1,P1, P2 , and P3 the
coefficients of temperature T (K) of poison's ratio ν for silicon nitride and stainless steel are listed in the following Table 1.
Table 1 Temperature dependent coefficients of poison's ratio ν for Si3N4/SUS304
Material Properties P-1
P0
P1
P2
P3
Si3N4
ν
0
0.2400
0
0
0
SUS304
ν
0
0.3262 -2.002e-4 3.797e-7
0
Length to thickness ratio (a/h) is 10 and a load ( q0 ) of 1 N/m2 was considered. A time step of 1e-6s was used. ΔT is 300 K.
The center deflection and time were non dimensionalized according to the following expressions:
W
wEm h3
q0 a 4

T t
Figure 4 Temporal evolution of central deflection
of clamped FGM plate under suddenly applied lateral
load in thermal environment
Em
ma2
(10)
Figure 5 Effect of temperature rise on the
dynamic response of clamped FGM plate
subjected to suddenly applied lateral load
We examined the effect of material composition on the transient response of clamped Si3N4/SUS304 square subjected to a
suddenly applied lateral load, under thermal environmental condition ΔT = 300 K. Figure 3 shows the deflections as
functions of varying power law index n. From the figure 3, it is known that as the power law index increases, means the
higher the bending rigidity is, the lower will be the peak value of deflections. From figure 4 shows the effect of thermal
environmental conditions ΔT = 0 and 300 K. It can be seen that deflections increase dramatically with increasing ΔT.
4. CONCLUSIONS
The dynamic analysis is carried out to investigate the performance of FGM plate due to mechanical load. In the present
work, a Finite element model of a FGM plate is developed. The nine-noded isoparametric element is used in the present FE
analysis. The element contains five mechanical displacement degrees of freedom per node. The governing dynamic
equilibrium equations for the FE analysis are formulated employing Hamilton's principle to the Lagrange energy functional
where the total energy consists of the kinetic energy, strain energy and the potential energy due to mechanical loads. The
static and free vibration problems are derived as the special cases of a general dynamic problem. Newmark's time
integration is used for transient response analysis. The present FE model is coded in MATLAB. Using these FE models the
analysis of FGM plate has been carried out. The good agreement of the present formulation is also validated by comparing
the numerical results with the published literatures. Now, based on the numerical analyses carried out for static and
dynamic problems, the following important conclusions are drawn for transient response analysis:
The response analysis exhibit that the higher the bending rigidity, the lower the magnitude of deflection. The amplitude of
vibration is maximum for the metallic plate and minimum for the ceramic plate. The transient response of graded plates is
intermediate to that of metal and ceramic plates. It is also clearly seen that the deflections in clamped case are much lower
corresponding to the simply supported case due to increases in stiffness. Under thermal environments, with the increase in
ΔT, deflections increase dramatically.
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
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