Meshless Analysis of Functionally Graded Plate with Different Algebraic Shear Deformation Theories

IJIRST 窶的nternational Journal for Innovative Research in Science & Technology| Volume 2 | Issue 08 | January 2016 ISSN (online): 2349-6010

Meshless Analysis of Functionally Graded Plate with Different Algebraic Shear Deformation Theories Rahul Kumar Department of Mechanical Engineering Birla Institute of Technology, Ranchi, 835215, India

D. Mahto Department of Mechanical Engineering Birla Institute of Technology, Ranchi, 835215, India

Jeeoot Singh Department of Mechanical Engineering Birla Institute of Technology, Ranchi, 835215, India

Abstract The flexural response of functionally graded materials (FGM) plates is investigated. The mathematical formulation of the plate subjected to different mechanical loading is presented utilizing different algebraic shear deformation theories. Multiquadric radial basis function (MQRBF) is used in present analysis. The effect of the volume fraction exponent, boundary conditions, side to thickness ratio on the flexural response of FGM plates under transverse load is studied.

Keywords: FGM, RBF, Shear deformation, Meshfree _______________________________________________________________________________________________________

I. INTRODUCTION FGM have received considerable attention in many engineering applications. They are inhomogeneous materials in which the material properties vary continuously in one or more directions. Most popular numerical techniques for analyzing plate problems are based on FEM and FDM. In the last few years a numerical tool that avoids the problem of mesh generation known as meshless method has gained momentum. It has received attention of many researchers primarily due to the flexibility in the construction of finite dimensional sub-spaces. Kansa (1990) introduced the concept of solving partial differential equations using radial basis functions (RBFs). Inverse multiquadric RBFs have been used to analyze composite plates by Xiang et al. (2009).

Singh et al (2014,2011,2011), Ferreira (2003) presented the analysis of laminated composite plates using multiquadric radial basis function. A review of meshless methods for laminated plated is presented by Liew et al. (2011). Qian and Batra (2004) and Dai et al. (2004) analysed the FGM plates using meshfree method. A radial basis functions meshless method utilizing first order shear deformation theory is used by Roque et al. (2010) to study the linear transient response of functionally graded plates and shells. Zhuo et al. (2009) presented a truly boundary only meshless method to solve Winkler plate bending problems under complex loads. In the present study, the flexural analysis of functionally graded plates using multiquadric radial basis function is presented. Functionally graded plates with clamped and simply supported boundary conditions and subjected to uniform transverse pressure are analyzed.

II. MATHEMATICAL FORMULATION The flexural behavior of functionally graded material (FGM) plate obtained due to various types theories is taken up for investigation. A rectangular plate having a, b edge length along x, y axes respectively and thickness h along z axis whose mid plane is coinciding with xy plane of the coordinate system is considered. The geometry of rectangular functionally graded material (FGM) plate in rectangular coordinate system is shown in Fig. 1.

Fig. 1: Geometry of rectangular FGM plate in rectangular coordinate system

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Meshless Analysis of Functionally Graded Plate with Different Algebraic Shear Deformation Theories (IJIRST/ Volume 2 / Issue 08 / 018)

The homogenization technique considered in this work is the law of mixtures, which provides the following elastic properties at each material layer. The top surface of the plate is ceramic rich and the bottom surface is metal rich. The volume fraction of the ceramic phase is given by  2z  h  Vc ( z )     2h 

n

(1.1)

where n is exponent governing the material properties along the thickness direction known as volume fraction exponent or grading index, The volume fraction of the metal phase is obtained by (1.2) Vm ( z )  1  Vc ( z ) The material property gradation through the thickness of the plate is assumed to have the following form n

 2z  h  E (z)  Ec  Em    Em  2h 

(1.3)

n

 2z  h    m  2h 

 (z)  c  m 

Here E and  denote the modulus of elasticity and density of FGM structure, while these parameters come with subscript m or c represent the material properties for pure metal and pure ceramic plate respectively., h is the thickness of the plate, E m and E c are the corresponding Young’s modulus of elasticity of metal and ceramic and z is the thickness coordinate

 h / 2  z 

h / 2 .

The displacement field at any point in the plate is expressed as: w0 ( x, y )

U  u0 ( x, y )  z

x w0 ( x, y )

V  v0 ( x, y )  z

y

 fi(z)  x ( x , y ) (2.1)  (2.3)

 fi(z)  y ( x , y )

W  w 0  x, y 

Where, fi(z)

 z  h2 z2       A m bartsum ian (1958)  f1  z    2  4 3            5z  4z 2    f2  z     1 K aczkow ski (1968), P anc (1975) and R eissner (1975)     4  3h 2    =  2       4z f3  z  z 1Levinson (1980), M urthy (1981) and R e dd y (1984)        2  3h           2   f4 z       1 5z z  Shim pi and P atel (2006)  4  2  3h    

transverse shear function

The parameters U, V and W are the in-plane and transverse displacements of the plate at any point (x, y, z) in x, y and z directions, respectively. u0, v0 and w0 are the displacements at mid plane of the plate at any point (x, y) in x, y and z directions, respectively. The functions  x and  y are the higher order rotations of the normal to the mid plane due to shear deformation about y and x axes, respectively. The governing differential equations of plate are obtained using Hamilton’s principle and expressed as. u 0 : v 0 :

N xx x N xy x

 

N xy y  N yy y

 x :

 M xx x M

2

f xx

x



f

 y :

M xy x

 qy  0

2

2

w 0 :

 qx  0



 M yy y

M

f xy

y

2

2

2

 M xy xy

 qz  0

(3.1  3.5)

f

 Qx  0

f



 M yy y

f

 Qy  0

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Meshless Analysis of Functionally Graded Plate with Different Algebraic Shear Deformation Theories (IJIRST/ Volume 2 / Issue 08 / 018)

The force and moment resultants in the plate are expressed as:  h/2 f ij

N ij , M ij , M





(  i j , z  i j , f ( z )  i j ) dz

 h/2  h/2 f Qx

,

f Qy





 h/2

(4.1  4.2 )

f  σ xz , σ yz    z  dz  

The plate stiffness coefficients are expressed as: n    2z  h  c m m    Q ij  Q ij     Q ij 2 h     h/2  h/2

Aij , B ij , D ij , E ij , Fij , H ij 

 



  2 2  1, z , z , f ( z ), z f ( z ), f ( z )   dz   

i, j = 1, 2, 6

(5.1)    2z  h  c m m    Q ij  Q ij     Q ij 2 h    h /2    h /2



Aij 



n



  f  z       z

  

2

   dz  

i, j = 4, 5 The boundary conditions for an arbitrary edge with simply supported and clamped edge conditions are

(5.2)

Simply Supported(SS):

x  0, a : N xx  0, v 0  0, w 0  0, M xx  0,  y  0 (6.1)

y  0, b : u 0  0, N yy  0, w 0  0,  x  0, M yy  0 Clamped(CC):

x  0, a : u 0  0, v 0  0, w 0  0,  x  0,  y  0 y  0, b : u 0  0, v 0  0, w 0  0,  x  0,  y  0

(6.2)

III. SOLUTION METHODOLOGY The governing differential equations (3.1 -3.5) are expressed in terms of displacement functions. Radial basis function based formulation works on the principle of interpolation of scattered data over entire domain. A 2D rectangular domain having NB boundary nodes and ND interior nodes is shown in Fig. 2 1 Interior Domain Nodes (NI) Boundary Nodes(NB)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2: An arbitrary two dimensional domain

The variable u 0 , v 0 , w 0 ,  x and  y can be interpolated in form of radial distance between nodes. The solution of the coupled non linear governing differential equations (3.1-3.5) is assumed in terms of multiquadric radial basis function (MQRBF) for nodes 1:N, as; N

u 0 , vo , wo ,  x ,  y 

 (

x u v w j , j , j , j

,

y j

) g



X X

j

,c



(7)

j 1

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108

Meshless Analysis of Functionally Graded Plate with Different Algebraic Shear Deformation Theories (IJIRST/ Volume 2 / Issue 08 / 018)

Where, N is total numbers of nodes which is equal to summation of boundary nodes NB and domain interior nodes ND. g



X X

j

,c



u

is radial basis function,





 j , j , j , jx , jy v

w

are unknown coefficients?

X X

j

is the radial distance

between two nodes. The RBF used in present solution methodology is Multiquadrics function and expressed as:



2

g  r c

Where, r  X  X

j

x  xj



2



 y yj



2



m

(8)

2

and m and c are shape parameters as in Singh and Shukla (2012).

IV. RESULT AND DISCUSSION The results obtained on the basis of previous discussions are presented here. Following material properties are taken : E c =151 GPa , ν=0.3 for ceramic; and E m =70 GPa , ν=0.3; for metal.

The deflection, stresses and the load parameters are non-dimensional zed as: _

w  wc / h,q   xx 

m ax(  xx )

q 0a

4

Emh ,  yy 

4

m ax(  yy )

,  xy 

(9)

m ax(  xy )

h h h Where, w c is transverse central deflection of the plate.

Convergence and validation study In order to show the accuracy and efficiency of the present solution methodology, convergence study of transverse central deflection ( w ) of the square FGM plate is carried out. For present analysis a suitable value of m=0.98, α = 0.23 and 0.68 are used for clamped and simply supported boundary conditions, respectively throughout the analysis on interior nodes. The convergence study for center deflection ( w ) of a square functionally graded plate for simply supported (SS) boundary conditions under uniformly distributed pressure is carried out and shown Table-1 and same is depicted in Fig.3. It can be seen that good convergence is achieved in all the cases for 13x13 or more number of nodes. Present result is in good agreement with the results obtained by Ferreira et al (2007). It can be seen that a 13 x 13 node is sufficient to yield accurate results and convergence is below 2 percent in all the cases, hence 13×13 node is used throughout the analysis.

Table-1 Convergence of central deflection 100w ̅ of a simply supported square FGM plate under uniform pressure (a/h=20, f3(z) ) Present Ferreira (2007) Nodes/n 0 1 2 0 1 2 7x7

0.02366

0.033706

0.036768

-

-

-

9x9

0.022898

0.032624

0.035584

-

-

-

11x11

0.022065

0.031453

0.034317

0.02

0.0297

0.0321

13x13

0.021688

0.030922

0.033735

0.0204

0.0305

0.0328

15x15

0.021446

0.030584

0.033371

0.0207

0.0308

0.0338

7x7

9x9

11x11

13x13

15x15

0.040

0.040

Normalised Central Deflection

0.038 0.036

0.036 0.034

0.032

0.032 0.030

n=0 n=1 n=2 n=5

0.028

0.028 0.026

0.024

0.024 0.022 7x7

9x9

11x11

13x13

15x15

Number of Nodes

Fig. 3: Convergence of central deflection of a simply supported square FGM plate under uniform pressure (a/h=20)

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109

Meshless Analysis of Functionally Graded Plate with Different Algebraic Shear Deformation Theories (IJIRST/ Volume 2 / Issue 08 / 018)

Numerical Examples: With an objective to compare the different theories a functionally graded material square plate is considered for the analysis. The normalized transverse displacements of the simply supported and clamped plates under uniform pressure q=1 are obtained. The comparison is shown in Table-2.

Table – 2 Comparison of the central deflection obtained due to f3(z) of a simply supported(SS) and Clamped (CC) square plate for different value of volume fraction exponent (n) and side to thickness ratio (a/h) a/h 4 10 20 50 100 n 0 0.02796 0.02244 0.02169 0.02149 0.02146 1 0.03950 0.03196 0.03092 0.03065 0.03061 2 0.04379 0.03496 0.03374 0.03341 0.03336 SS 5 0.04851 0.03792 0.03644 0.03606 0.03600 10 0.05167 0.04039 0.03882 0.03841 0.03836 10000 0.06029 0.04840 0.04677 0.04634 0.04628 0 0.00943 0.00775 0.00731 0.00715 0.00712 1 0.01297 0.01076 0.01018 0.00997 0.00994 2 0.01418 0.01172 0.01106 0.01082 0.01078 CC 5 0.01574 0.01287 0.01209 0.01179 0.01175 10 0.01699 0.01386 0.01300 0.01268 0.01263 10000 0.02033 0.01672 0.01577 0.01542 0.01536 The results for central deflection is obtained and shown in Table-3 and same is depicted in Fig.4 to compare different theories taken here. It is observed that theory 1 and 3 gives similar results for thick to thin plates.

Table-3 Comparison of different theories on the central deflection (n=5) a/h f1(z) f2(z) f3(z) f4(z) 4 0.048510 0.045910 0.048510 0.045960 10 0.037920 0.037490 0.037920 0.037490 SS 20 0.036440 0.036340 0.036440 0.036340 50 0.036060 0.036040 0.036060 0.036040 100 0.036000 0.036000 0.036000 0.036000 4 0.015740 0.015250 0.015740 0.015260 10 0.012870 0.012670 0.012870 0.012670 CC 20 0.012090 0.012020 0.012090 0.012020 50 0.011790 0.011780 0.011790 0.011780 100 0.011750 0.011740 0.011750 0.011740 The results of theory 2 and 4 are also similar. But it can be concluded that for thick plates theory 2&4 under predicts the deformation as compared to that of theory 1&3. However for thin plates results obtained by all the theories taken here are very close.

0.050

Central Normalised Deflection

0.048 0.046 0.044

f1(z) f2(z) f3(z) f4(z)

0.042 0.040 0.038 0.036 0

10

20

30

40

50

a/h ratio

Fig. 4: Comparison of different theories on the central deflection of simply supported square FGM plate (n=5) Stresses and deflections of thin to thick FGM plates for under uniform pressure are obtained for simply supported and clamped FGM plate and is shown in Table-4.

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Meshless Analysis of Functionally Graded Plate with Different Algebraic Shear Deformation Theories (IJIRST/ Volume 2 / Issue 08 / 018)

Table-4 Comparison of the central deflection and stresses square plate (n=5, f3(z)) Simply Supported Clamped (w) 0.0485 0.0157 σx 6.7329 1.0734 4 σy 4.6064 0.2306 σxy 3.3333 0.7267 (w) 0.0360 0.0117 σx 0.1904 0.0329 100 σy 0.1277 0.0101 σxy 0.1030 0.0225

V. CONCLUSIONS The flexural response of functionally graded plate (FGP) under different theories are obtained using MQRBF. The results presented herein show the applicability of present solution methodology. It can be concluded that for thick plates theory 2&4 under predicts the deformation as compared to that of theory 1&3. However for thin plates results obtained by all the theories taken here are very close.

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