257
ON THE USE OF MESHLESS METHOD FOR FREE VIBRATION ANALYSIS OF CIRCULAR FGM PLATE HAVING VARIABLE
THICKNESS UNDER AXISYMMETRIC CONDITION
Abazar Shamekhi
Department of Mechanical Engineering, University of Alberta, Edmonton AB T6G 2G8 Canada
Email: [email protected] Tel: 1 780.492.0802
A
BSTRACTIn this work the free vibration analysis of circular plate having variable thickness made of functionally-graded material is studied. Numerical analysis has been done for either simply supported or clamped circular FGM plates.
Dynamic equations have been obtained using energy method based on Love-Kichhoff hypothesis and Sander's non- linear strain-displacement relation for thin plates. Mesh free method is used to determine the natural frequencies.
The results obtained show good agreement with known analytical data. The effects of thickness variation and Poisson's ratio are studied by calculating the natural frequencies. These effects are found to be different for simply supported and clamped plates.
Keywords: Vibration, FGM, Meshless Method.
NOMENCLATURE: a: Plate radius D: Flexural rigidity E: Modulus of elasticity
e: Neutral plane location in Z coordinate k: The volume fraction exponent n: Number of nodes
P: Vector of monomials p: Monomials
r: Radial Position T: Kinetic energy t: Thickness of the plate U: Strain energy
W: Generalized displacement vector
Ws: Support domain generalized displacement vector w: Deflection of plate in z coordinate
Z: Coordinate axis across the plate thickness originated from middle plane z: Coordinate axis across the plate thickness originated from neutral plane GREEK LETTERS:
: Coefficients of monomials
: Coefficient that shows thickness variation
: Mass coefficient
: Slope
: Poisson's ratio
: Density
: Stress
: Matrix of shape functions
: Shape function
: Problem domain
s: Support domain
: Natural frequency258 SUBSCRIPTS:
c: Ceramic i: Node Number m: Metal
Q: Assumed point (often Gaussian point for integration) r: Radial component of polar coordinate
s: Support domain
0: Value of a variable in center of the plate
: Tangential component of polar coordinate SUPERSCRIPTS:T: Transpose of a matrix (1): First derivation ABBREVIATIONS:
FGM: Functionally graded material MFree: Mesh Free
EFG: Element Free Galerkin PIM: Point Interpolation Method PDE: Partial Differential Equation 1. INTRODUCTION
Functionally-graded materials (FGM) were first introduced in 1984 by a group of scientists in Japan. FGMs are used in aircrafts, space shuttles and etc. as thermal resistant materials. FGMs are composite materials in which the material properties vary continuously from one face to the other. The most important advantage of FGMs is the ability to protect their surrounded materials under high temperature gradients without losing their structural integrity [1]. These materials are usually made of a mixture of ceramics and metals. The ceramic component has low thermal conductivity and the metal component prevents fracture of material due to high temperature gradients. FGMs are manufactured in different shapes and configurations (for more details the reader is referred to the literature [2, 3]).
Vibration analysis of FGM plates has been investigated by several authors. Ng et al. [4] presented parametric resonance of functionally graded rectangular plates under harmonic in-plane loading. They investigated the effects of configurations of the constituent materials on the dynamic behavior of the plates. Han and Liu [5] analyzed the SH waves propagating in FGM plates by numerical methods. They employed inversed Fourier integration to obtain displacements and stress in the frequency domain and time domain. They proposed a simple integral technique for evaluating the modified Bessel function with complex valued order. Chakraborty and Gopalakrishnan [6] studied the wave propagation behavior in functionally graded beam subjected to high frequency impulse loading by spectral finite element. They investigated dependence of cut-off frequency and maximum stress gradient on material properties. Berezovski et al. [7] studied the propagation of stress waves in functionally graded materials by means of the composite wave-propagation algorithm. They used two distinct models of FGMs. They found that these two models show significant difference in stress wave characteristic.
Circular plates are used in many engineering structures. In many cases variable thickness is used in order to save weight and improve structural characteristics. For example, in buckling loads, a circular plate having variable thickness has more strength than a uniform circular plate having similar weight [8]. Yoshihiro and Junichi and Hiroyuki formulated a transient plane-stress thermoelastic problem in a nonhomogeneous hollow circular plate of variable thickness. They presented both analytical and numerical solutions for thermal stresses in a nonhomogeneous hollow circular plate subjected to unaxisymmetic heating on the boundary surfaces, which has the plate thickness, Young's modulus and thermal conductivity in forms of different power laws of the radial coordinate [9]. Goering and McClain discussed recent advancements in the fabrication of three dimensionally woven 'Pi' inserts. They presented two main types of perform; 1) preforms that have variable cross sections (i.e. variable thickness and/or variable clevis width), and 2) preforms that can be formed to curved shapes through the use of stretch broken carbon fiber [10].
In many engineering applications the Poisson's ratio is not constant. Roberts and Garboczi studied elastic properties of model porous ceramics. They have also generated data that shows the dependence of Poisson's ratio on porosity and the solid Poisson's ratio. They concluded that at sufficiently high porosities, the Poisson's ratio converges to a fixed non-zero value [11].
Meshless methods have been used for analysis of plates by some other authors. Krysl and Belystschko [12, 13]
developed the element free Galerkin (EFG) method to analyze thin plates and shells under static loads. They used
259
Lagrange multiplier for applying boundary conditions. Ouatouati and Johnson [14] extended EFG method for modal analysis of plates. Vibration and static behavior of thin homogenous and composite laminate plates with EFG method has been investigated by Liu, G. R. and Chen, X. [15,16].
In most papers mentioned above, the thickness of the plate was assumed to be constant across the plate. The goal of this paper is to derive the general dynamic equation for functionally-graded circular plates having variable thicknesses. Natural frequencies and mode shapes have been obtained by Meshless method. Galerkin weak form formulation is used for solving the problem, and point interpolation method (PIM) is employed for constructing the shape functions. The effect of thickness variation of the plate on the natural frequencies is investigated in either simply supported or clamped FGM circular plate. In special cases the results are verified with known analytical data in literature.
2. BASIC EQUATIONS
For a solid circular plate of radius a and variable thickness h, the material properties of the FGM plate, (e.g. the modulus of elasticity, E and density,
) is assumed to be a function of the volume fraction of the constituent materials. Denoting the coordinate axis across the plate thickness by Z (measured from the middle of the plate thickness), the relationships between E,
and Z for FGM plate made of ceramic and metal are as follows [17].
c m
km 2
1 t E Z E E ) Z ( E
E
,
2 Z t 2 t
, (1)
c m
km 2
1 t
z Z
( ) ,
2 Z t 2 t
(2)
where Ec ,
cand Em ,
mare the modules of elasticity and density of the ceramic and metal component, respectively, k is the volume fraction exponent (which can take values greater than or equal to zero), and t is the thickness of the plate. When a plate is subjected to pure bending, there is a plane with stress equal to zero. That plane is known as the neutral plane. In FGM plates, the neutral plane is not coincided with the mid-plane. In this paper stress and strain across the thickness of the plate are measured from the neutral plane because this coordinate transformation simplifies the equations.Figure 1 shows the neutral plane which is located in position e from the mid-plane. Denoting the coordinate axis across the plate thickness by z, (measured from the neutral plane), the elasticity modulus and density equations change to the following form:
c m
km 2
1 t
e E z E E ) z ( E
E
, e
2 z t 2 e
t
, (3)
c m
km 2
1 t
e
z z
( ) , e
2 z t 2 e
t
. (4)
Mid-plane Neutral plane
Figure 1. Neutral Plane location in the FGM plate
The Poisson's ratio,, is assumed to be constant across the plate thickness. The relationship between stress and deflection of the plate in axsisymmetric bending is as follows [18]:
dr
dw r dr
w d z z E
r
2 22
1 ).
( , (5)
2 1 22
1 dr
w d dr dw r z ).
z (
E
, (6)
where r and are polar components of stress, and w is the deflection of plate in z coordinate. It is assumed that the plane is subjected to pure bending [14], thus, the summation of all infinitesimal forces must be equal to zero,
260
1 0
1 2
2 2
2 2 2
2 2
2
2
dr dz w d dr dw z ).
z ( dz E
dr dw r dr
w d z ).
z ( E
t e
t e t e
t e
. (7)
Equation (7) is true for any section of the plate, therefore Eq. (7) can be simplified to the following form.
2e t
e 2 t
0 zdz ).
z (
E . (8)
Substituting Eq. (3) into Eq. (8), and after some mathematical manipulations, the position of the neutral plane can be obtained as:
m m c m
m c
E E E 1 k 1 1
2 k 2
1 2 k
1 E
E E t
e . (9)
The strain energy of circular plates with axisymmetric pure bending is as follows [18]:
dr rdr w d dr dw r 1 2 dr dw r 1 dr
w D d
U 2
2 2 a
0 2 2
, (10)where D is flexural rigidity of the plate which is obtained as follows:
t e
t e
m c
m k
t e k
t e k
E E e t
e t E t z dz
).
z ( D E
2
2
2
2 3 3
3 2
2 2
1 4 1 2
2 1 2
3 1 1
2 2
1 3
1
1
. (11)
The maximum kinetic energy can be determined by the following equation [19]:
a
0 2 2
max 2 w rdr
T
. (12)
Natural frequencies can be obtained from following equation [19]:
max
max T
U
. (13)
Therefore, it can be written:
a
0 2 2 2
2 2 a
0 2 2
rdr w 2
dr rdr w d dr dw r 1 2 dr dw r 1 dr
w
D d
. (14)
Natural frequencies can be obtained by:
a
0 2
2 2 2
a
0 2 2
2
rdr w 2
dr rdr w d dr dw r 1 2 dr dw r 1 dr
w D d
. (15)
To calculate natural frequencies, Eq. (15) must be minimized [19]. The technique of calculus of variations will be used for this purpose.
3. NUMERICAL ANALYSIS
In general there is no analytical closed-form solution for free vibration analysis of variable thickness plates.
Therefore, to obtain natural frequencies and mode shapes, numerical Mesh Free method is employed. Mesh free method (known as MFree method) is a new method for solving boundary value and initial value PDEs (Partial differential equations). This section discusses this method in brief.
3.1.SHAPE FUNCTION CONSTRUCTION
There are many techniques for constructing shape functions in Meshless method. In this paper point interpolating method (PIM) is employed for this purpose [20]. Using PIM shape functions has some advantages compared to other Meshless shape functions.
1. PIM shape functions possess the Kronecker delta function property. Therefore, applying essential boundary conditions is very easy in this method.
261
2. PIM shape functions satisfy a certain order of consistency [20].
3. Calculation of PIM shape functions' derivatives is very simple.
4. There is no need for weight functions in construction of PIM shape functions.
Although PIM shape functions have above excellent advantages, they have some disadvantages that restrict their applications in many engineering problems. The approximation that uses PIM shape functions can be discontinuous over problem domain. Furthermore, PIM shape function moment matrix can be ill conditioned during the interpolation procedure. Ill conditioning of PIM moment matrix can be treated by using some techniques such as coordinate transformation or matrix triangularization algorithm [20].
In this work, the weak form of differential equation for pure bending is second-order. Therefore, for any nodes we must save deflection and slope data. Boundary conditions are axisymmetric thus the solution is reduced to just one dimension. Figure2 shows the nodes included in the problem domain. In this figure, ri is the radial distance of the ith node from the center of the plate, rQ is the position of an assumed point (often a Gaussian point for the integration),
s is support domain [20]. The deflection w is approximated using PIM shape functions that are constructed using a set of nodes included in the 1D support domain sof a point of interest rQ as shown in Figure2. Therefore it can be written:
2n
1 i
Q T
Q i
Q) p(r). (r ) P (r). (r )
r , r (
W , (16)
where pi(r) are monomials of r, n is the number of nodes in the support domain of rQ, and
i( r
Q)
is the coefficient for pi(r). Matrix PT(r) in Eq. (1) is as follows [20]:} ,..., , , , { )
( 2 3 2n1
T r 1r r r r
P . (17)
ri sNodes rQ Neutral plane
Figure 2. A schematic diagram showing the nodes included in the problem domain
The derivatives of the field variable of the deflection can be obtained simply by using Eq. (16). The first derivative of deflection is as follows [20]:
2n
1 i
Q T 1 i Q 1
i Q
1 r r p r r P r r
W
r) ( , ) ( ). ( ) { ( )} . ( )
( () () ()
, (18)
where,
T
r n
n r
r r
P(1)( ){0,1,2 ,3 2,...,(2 1) 2 2} . (19) The coefficient
i in Eq. (16) and Eq. (18) can be obtained by using Eq. (16) and Eq. (18). At node i we have equation:
[ ( )] .
).
( )
( ).
(
)
( T
i 1
i T n
2
1 i
i i i n 2
1 i
i i i
i i
i P r
r P dr
r dp
r w p
W , (20)
where wi and i are nodal values of w and at r=ri. In matrix form, Eq. (20) can be written as [20]:
. P
Ws Q , (21)
where Ws and PQ are the generalized displacement vector and moment of the node in the support domain, respectively. That is [20]:
1 1 2 2 n n
s w w w
W , , , ,..., , , (22)
)]
( ), ( ),..., ( ), ( ), ( ), (
[ 1 (1) 1 2 (1) 2 n (1) n
T
Q p r p r p r p r p r p r
P , (23)
where
can be obtained from Eq. (21). We have:262
s 1
QW
P
. (24)
Substituting Eq. (24) into Eq. (16), we obtain:
s T(r).W )
r (
w , (25)
where is the matrix of the shape function in the following form [20]:
( ), ( ), ( ),..., ( )
)
(r 1 r 2 r 3 r 2n r
T
. (26)
3.2.SOLUTION METHOD
There are many solution techniques for solving partial differential equations. In this paper Galerkin weak form formulation is used to solve partial differential equation with the advantage that imposing essential boundary condition is very simple with PIM shape function. According to Eq. (15), it can be written:
a
0 2 a
0 2 2
2
rdr w 2
rdr dr dw r 1 dr2
2w d D D
D D dr dw dr
w d
.
. ,
. (27)
Substituting Eq. (25) in Eq. (27) we obtain:
~} .{
) ) ( . ) ( .
~} {
~} .{
) (
) ( ) .
, ( ) . (
~}
{
.
W rdr r r 2 W
W rdr dr
r d r 1
dr2 2 r d D D
D D dr
r d dr
r W d
a
0 T T
a
0
T 2 T 2 T
2
, (28)
~} ].{
.[
~} {
~} ].{
.[
~} {
W B W
W K W
T T 2
, (29)
where {
W ~
} is the global generalized displacement vector that contains all nodes’ displacements in problem domain. Furthermore,
a
0
T 2
T 2
rdr dr
r d r 1
dr2 2 r d D D
D D dr
r d dr
r K d
) (
) ( ) .
, ( ) ] (
[
.
, (30)
a
0
T r rdr
r 2
B] ( ) . ( )
[ . (31)
To minimize
2, we must have:} 0 W~ {
2
. (32)
Substituting Eq. (29) in Eq. (32), we have:
2 T
T T 2
W B W
W K W W B 2 W B W W K 2
W ~})
].{
.[
~} ({
~}) ].{
~}.[
~}.({
].{
[
~}) ].{
.[
~} .({
~} ].{
[
~} {
. (33)
According to Eq. (29) (after some manipulations), we should have:
} 0 ].{~ [
~} {
~} ]{
[
~} ].{
[ 2
W B W
W B W
K
T , (34)
~} {
~} ]{
.[
]
[B1 K W 2 W . (35)
Equation (35) is an eigenvalue problem. It can easily be solved using known numerical procedures. PIM shape functions permit us to apply boundary conditions easily on boundary nodes (like finite elements methods). After solving the eigenvalue problem, natural frequencies and mode shapes can be obtained straightforwardly.
263 4. RESULTS AND DISCUSSION
To solve the dynamic equation with Meshless method, a new code was developed in this work. Power method is used for obtaining natural frequencies and mode shapes. In this work the thickness of the plate is assumed to vary linearly from the center of the plate. Therefore, it can be calculated from the following equation:
a
. r 1 t
t 0 , (36)
where t is thickness at position r from the center, t0 is the thickness of the center of the plate, a is plate radius and is a coefficient showing thickness variation. Denoting mass coefficient of plate by
, it is defined by following equation (Z is measured from the mid-plane):
2
t
2 t
dZ
.
, (37)
Tables 1,2 and Figs. 3,4 show minimum dimensionless natural frequency for FGM circular plate with variable thickness for different values of Poisson’s ratios (D0 and
0 are flexural rigidity and mass coefficient of the center of the plate)0 1 2 3 4 5 6 7 8 9
-0.99 -0.8 -0.61 -0.42 -0.23 -0.04 0.15 0.34 0.53 0.72 0.91
v=0.0 v=0.2 v=0.4
Figure 3. First dimensionless natural frequency of simply supported circular FGM plate with variable thickness and different values of Poisson’s ratios
0 0 2 11
D a
264
0 2 4 6 8 10 12 14 16 18 20
-0.99 -0.79 -0.59 -0.39 -0.19 0.01 0.21 0.41 0.61 0.81
v=0.0 v=0.2 v=0.4
Figure 4. First dimensionless natural frequency of clamped circular FGM plate with variable thickness and different values of Poisson’s ratios
Table 1. The dimensionless natural frequency (
0 0 2 11
D a
), in simply supported plate.
The dimensionless natural frequency (0 0 2 11
D a
)
0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.0 3.389 3.650 3.913 4.177 4.443 4.712 4.983 5.257 5.533 0.1 3.545 3.814 4.081 4.349 4.619 4.890 5.162 5.437 5.713 0.2 3.694 3.968 4.240 4.511 4.782 5.054 5.326 5.600 5.875 0.3 3.836 4.115 4.390 4.663 4.935 5.206 5.477 5.748 6.020 0.4 3.971 4.255 4.533 4.807 5.078 5.347 5.615 5.883 6.150 0.5 4.102 4.389 4.669 4.943 5.212 5.479 5.743 6.005 6.267
Table 2. The dimensionless natural frequency (
0 0 2 11
D a
), in clamped plate.
The dimensionless natural frequency (0 0 2 11
D a
)
0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.0 6.760 7.617 8.478 9.345 10.215 11.090 11.968 12.850 13.735 0.1 6.829 7.671 8.516 9.364 10.215 11.070 11.927 12.788 13.651 0.2 6.898 7.725 8.553 9.383 10.215 11.050 11.887 12.726 13.567 0.3 6.965 7.778 8.590 9.402 10.215 11.030 11.845 12.663 13.481 0.4 7.032 7.830 8.627 9.421 10.215 11.009 11.804 12.599 13.395 0.5 7.097 7.883 8.663 9.440 10.215 10.989 11.762 12.535 13.308
In simply supported plate dimensionless natural frequency increases due to an increase in and , but, in clamped plate the situation appears to be quite different. For negative the dimensionless natural frequency increases by an
0 0 2 11
D a
265
increase in , whereas for positive the dimensionless natural frequency decreases when Poisson’s ratio is increased. Figure 5. shows dimensionless flexural rigidity of FGM plate for different volume fraction exponents and different two components elasticity modulus ratio. For a solid circular homogenous plate having constant thickness, the natural frequencies are as follows [19]:
0 0 2 2 mn mn
D a
1
, (38)where mn are eigenvalues that can be obtained from following equation:
0 ) ( I ).
( J ) ( J ).
(
Im1 m2 m1 m2 , m=1, 2, 3, … (39) where Im and Jm are Bessel functions. This equation can be solved numerically. Therefore, the first eigenvalue is as follows:
2158 10 D
a 2
11 0
0 2
11 .
. (40)
Tables 3. shows result validation for homogenous plate with uniform thicknesses. It shows that the results obtained in this work completely agree with other analytical data. Figures 6,7. show first vibration mode shape of simply supported and clamped plate with different .
0 0.2 0.4 0.6 0.8 1 1.2
0 1 2 3 4 5 6 7 8 9 10
E=1.0 E=1.5 E=2.5 E=6.0
k
Figure 5. Flexural rigidity of FGM plate using different the volume fraction exponent and different E (
m c
E E E )
0 0.2 0.4 0.6 0.8 1 1.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
=-0.9 =-0.5 =0.0 =2.0 =4.0
a r
Figure 6. First mode shape of vibration (clamped plate)
3 2
. 1 12
t E
D
c
Wmax
W
266
0 0.2 0.4 0.6 0.8 1 1.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
=-0.9 =-0.5 =0.0 =2.0 =4.0
a r
Figure 7. First mode shape of vibration (simply supported plate)
Table 3. Result validation (
0 0 2 11
D a
) for homogenous circular plate with uniform thickness (0.3)
The dimensionless natural frequency (
0 0 2 11
D a
)
Clamped
This work 10.215
Reference [19] 10.215
Figure 8. demonstrates the effect of the number of nodes on the normalized error. The larger the number of nodes, the higher the accuracy of the results. For the number of nodes greater than 64 the accuracy decreases slowly. Figure 9. shows the support domain scale effect on the normalized error. Support domain scale is defined as the half of support domain length over averaged distance between two neighboring nodes. According to Figure 2 the support domain scale can be defined as follows:
ave s
s 2L
, (35)
where s,sand Lave are the support domain scale, the support domain length and the average distance between two neighboring nodes, respectively. The vibration analysis is very sensitive to the support domain scale. For support domain scale between 0.95 and 1.6, the normalized error is less than 2 percent. To obtain K and B matrices discussed above, a background mesh with Gauss quadrature integration method is used. Figure 10 shows the effect of number of Gaussian points on the normalized error. The vibration analysis is not very sensitive to the size of background meshes. To have an accurate solution, the number of Gaussian integration points should increase with the increase of the size of background meshes. According to these results, number of Gaussian points between 2 and 4 is recommended. Using high Gaussian points increases CPU time significantly.
Wmax
W
267
1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00
0 20 40 60 80 100 120
Normalized Error
Number of Nodes
Figure 8. The effect of number of nodes on accuracy
1.0E-08
1.0E-07 1.0E-06
1.0E-05
1.0E-04 1.0E-03
1.0E-02
1.0E-01
1.0E+00
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Normalized Error
Support domain coefficient
Figure 9. The effect of support domain scale on accuracy
1.0E-10 1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00
0 2 4 6 8 10
Normalized Error
Number of Gaussian Points
Figure 10. The effect of number of Gaussian point on accuracy
268 5. CONCLUSION
A general dynamic equation was obtained for functionally-graded circular plates having variable thickness. Natural frequencies and mode shapes have been obtained by Meshless method. Galerkin weak form formulation is used for solving the problem, and point interpolation method (PIM) is employed for constructing the shape functions. The effect of thickness variation and Poisson's ratio is investigated by calculating natural frequencies. These effects are not the same for both simply supported and clamped plates. The results are validated with known analytical data. In numerical solution the effect of number of nodes, support domain scale, and number of Gaussian points are discussed. It was concluded that the vibration analysis is very sensitive to support domain scale.
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