Vol-7, Special Issue-Number5-July, 2016, pp1185-1195 http://www.bipublication.com
Research Article
Wave stress propagation in a rectangular FGM plate rest on
winkler foundation
Ali Reza Fallah1, Ali Ghorbanpour Arani2 And Reza Shahveh3
1
Student of Mechanic ,University of Azad, Jasb, Islamic Republic of Iran
2Institute of Nanoscience and Nanotechnology,
University of Kashan And Kashan, Islamic Republic of Iran
3
P.H.D Student of Mechanic,
University of Science and Research of Tehran, Tehran, Islamic Republic of Iran
ABSTRACT
The wave propagation of a functionally graded plate is studied using the first-order shear deformation plate theory. Material graded properties are assumed in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by solving an eigenvalue problem. Numerical examples show that the characteristics of wave propagation in the functionally graded plate are relates to the volume fraction index of the functionally graded plate. The influences of the volume fraction distributions on wave propagation of functionally graded plate are discussed in detail. The results carried out can be used in the ultrasonic inspection techniques and structural health monitoring.
Key words: Wave propagation, functionally graded plate, first-order shear deformation, winkler foundation
1. INTRODUCTION
The concept of functionally graded materials (FGMs) were the first introduced in 1984 by a group of material scientists in Japan, as ultrahigh temperature resistant materials for aircraft, space vehicles and other engineering applications. Functionally graded materials (FGMs) are new composite materials in which the micro-structural details are spatially varied through non-uniform distribution of the reinforcement phase. This is achieved by using reinforcement with different properties, sizes and shapes, as well as by interchanging the role of reinforcement and matrix phase in a continuous manner. The result is a
microstructure that produces continuous or smooth change on thermal and mechanical properties at the macroscopic or continuum level [1,2]. Now, FGMs are developed for general use as structural components in extremely high temperature environments. Therefore, it is important to study the wave propagation of functionally graded materials structures in terms of non-destructive evaluation and material characterization.
Chen et al. [3] studied the dispersion behavior of waves in a functionally graded plate with material properties varying along the thickness direction. Han and Liu [4] investigated SH waves in FGM plates, where the material property variation was assumed to be a piecewise quadratic function in the thickness direction. Li et al.[5] used the WKB method to investigate the features of Love waves in a layered functionally graded piezoelectric. Chiu and Erdogan [6] studied the one-dimensional wave propagation in a functionally graded elastic medium. Zhang and Batra [7] used the modified
smoothed particle hydrodynamics (MSPH)
method to study the propagation of waves in functionally graded materials. Han et al. [8] proposed an analytical-numerical method for analyzing the wave characteristics in FGM cylinders. Han et al. [9] also proposed a numerical method to study the transient wave in FGM plates excited by impact loads. Considering the thermal effects, Chakraborty and Gopalakrishnan [10] used the spectral finite element method to analyse the wave propagation behavior in a functionally graded beam subjected to high frequency impulse loading based on the first-order shear deformation theory, where the material properties are temperature-independent, and graded in the
thickness direction only. Chakraborty and
Gopalakrishnan [11] studied the wave propagation behavior in a functionally graded beam subjected to high frequency loading by using a new beam finite element method based on the first-order shear deformation theory. Bahtin and Eslani [12] analyzed the coupled thermo elastic response of a functionally graded circular cylindrical shell, where the material properties were thickness direction according to a volume fraction power law distribution only. Sun and Luo [13,14] investigated the wave propagation and transient response of a FGM plate under a point impact load. Sun and Luo [15] also studied the wave propagation and dynamic response of rectangular
functionally graded material plates with
completed clamped supports under impulsive load. Yu et al. [16] used the Legendre orthogonal
polynomial series expansion approach to
investigate characteristics of guided waves in graded spherical curved plates. Elmaimouni et al. [17] proposed a numerical method to analyze the guided wave propagation in an infinite cylinder composed of functionally graded materials (FGM). Ghorbanpour et al. [18] a radially piezoelectric functionally graded rotating disk is investigated by the analytical solution. The variation of material properties is assumed to follow a power law along the radial direction of the disk. Two resulting fully coupled differential equations in terms of the displacement and electric potential are solved directly. Numerical results for different profiles of inhomogeneity are also graphically displayed. Yu et al. [19] also studied guided thermoelastic waves in FGM plates in the context of the Green–Lindsay (GL) generalized thermoelastic theories, but they did not take into
account temperature-dependent material
properties.
used in the ultrasonic inspection techniques and structural health monitoring.
2. PROPERTIES OF THE FGM
CONSTITUENT MATERIALS
An FGM plate of thickness (h) is considered here. The materials in top and bottom surfaces of the plate are ceramic and metal, respectively. The material properties (P) of FGMs are a function of the material properties and volume fractions of the all constituent materials which can be expressed as [20]
(1)
In Eq. (1) and are the ceramic and metal
volume fractions, respectively. The sum of volume fractions of the all constituent materials must be unity as follow
(2)
It is assumed that the material composition in an FGM plate varies continuously along the thickness direction only and distribute according to a power law as [22]
(3)
Where N denotes the power law index which takes values greater than or equal to zero. When N=0, the plate is homogeneous. It is assumed that the
effective Young’s modulus E, Poisson’s ratio ν
and mass density ρ of an FGM plate.
Using Eqs. (1), (3) they can be written as
(4)
An FGM plate defined in figure 1.
2.1 FUNDAMENTAL EQUATIONS
Considering the effects of rotary inertia and using the first-order shear deformation plate theory [25– 27], the displacements are assumed follows:
(5)
Where , and w are the total displacements and
are the midplane displacements in the
x,y and z directions, respectively, and are
the rotations along the x and y axis, respectively.
(6)
Using Eqs (5) and (6), the stress-strain relationships of the FGM plate can be written as
(7)
Where
(8)
The force and moment resultants of the FGM plate can be obtained by integrating Eq. (7) over the thickness, and are written as
in which ( ) denote the total in-plane
force resultants, ( ) denote the moment
resultants, ( ) denote the shear forces.
Substituting Eqs. (6) and (7) into Eq. (9) , we obtain:
(10)
(11)
Using the Hamilton’s principle and considering the effects of transverse shear deformation and rotary inertia, the governing equations of the FGM plate are given by
(12)
Winkler external force is applied to the substrate can be obtained from the following formula:
(13)
(14 )
In the above equation is the Winkler spring
factor.
UsingEqs. (10), (14) can be expressed in terms of the displacements
(1 5)
The Eq. (15) are the governing equations of the FGM plate in terms of the displacements.
3. DISPERSION RELATIONS
We assume solutions for , , , and
representing propagating waves in the plane
with the form
where and are the coefficients
of the wave amplitude, and are the wave
numbers of wave propagation along x-axis and y-axis directions respectively, is the frequency. Substituting Eq. (16) into Eq. (15), we obtain:
(1 7)
Eq. (17) can be further written in matrix form as
(18)
Where
(1 9)
and (i = 1, 2, 3, 4, 5; j = 1, 2, 3, 4, 5) are
given in Appendix A.
The dispersion relations of wave propagation in the functionally graded plate are given by
(20)
Assuming , the roots of Eq. (20) can
be expressed as
(2 1) The phase velocity of wave propagation in the functionally graded plate can be expressed as
(22)
The group velocity of wave propagation in the functionally graded plate can be expressed as
(23)
4. NUMERICAL RESULTS AND
DISCUSSION
An (Aluminum-Zirconia) functionally graded material plate is considered here. The thickness of the functionally graded plate is 0.01 m. The Young’s modulus E, density q, Poisson’s ratio of these materials are listed in Table 1, which are taken from reference [30]. The eigenvalues problem in Section 4 can be solved using Mathematica.
Fig. 2 shows the dispersion curves of the different functionally graded plates. It can be seen that the dispersion curves of the functionally graded plates are influenced by volume fraction distributions.
For the same , the frequency of the wave
propagation in the functionally graded plate decreases as the volume fraction index N increases. And, the frequency of the wave
propagation in the homogeneous plate ( ) is
the maximum among those of all functionally graded plates.
Figs. 3 and 4 show, respectively, the phase velocity and group velocity curves of the different functionally graded plates. It is seen that the phase velocity and group velocity of the wave propagation in the functionally graded plate decreases as volume fraction index N increases for the same . The phase velocity and group velocity
for M0 and M2 modes of the plate ( ) is a
constant, but it is not a constant for the plate
( ). The phase velocity and group velocity of
velocity of the wave propagation in the functionally graded plate.
5. CONCLUSIONS
Considering the effects of transverse shear deformation and rotary inertia, the wave propagation of functionally graded plate is studied. Material graded properties are assumed in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The analytic dispersion relation of the functionally graded plate is obtained by solving an eigenvalue problem. The influences of the volume fraction distributions on wave propagation of functionally graded plate are analyzed. From this research, the following concluding remarks can be made:
(1) The influence of the volume fraction distributions on wave propagation in the functionally graded plate is significant. When the wave number ( ) is given, the frequency of the wave propagation in the functionally graded plate decrease with increasing the volume fraction index N. The frequency of the wave propagation in the homogeneous plate (N=0) is the maximum among those of all functionally graded plates. (2) The influence of the heterogeneity of functionally graded materials on the wave propagation in the functionally graded plate is great. When the wave number ( ) is given, the phase velocity and group velocity of the wave propagation in the functionally graded plate decrease as volume fraction index N increases. The phase velocity and group velocity of the wave propagation in the homogeneous plate (N= 0) is the maximum among those of all functionally graded plates.
(3) Witch increases n to a specified value (increase of metal phase) in the first mode, the second and third phase velocity decreases, which shows the ceramic phase, signal permit easier runs.
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Figure (1): FGM plate rest on winkler foundation
(c) mode (d) mode
Figure (2): The dispersion curves of the different functionally graded plates
(c) mode (d) mode
Figure (3): The phase velocity curves of the different functionally graded plates
(c) mode
(d) mode
