Non-linear analysis of porous elastically supported FGM plate under various loading

Journal Pre-proofs Non-linear analysis of porous elastically supported FGM plate under various loading Rahul Kumar, Achchhe Lal, B.N. Singh, Jeeoot Singh PII: DOI: Reference:

S0263-8223(19)32600-5 https://doi.org/10.1016/j.compstruct.2019.111721 COST 111721

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Composite Structures

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9 July 2019 8 October 2019 21 November 2019

Please cite this article as: Kumar, R., Lal, A., Singh, B.N., Singh, J., Non-linear analysis of porous elastically supported FGM plate under various loading, Composite Structures (2019), doi: https://doi.org/10.1016/j.compstruct. 2019.111721

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Non-linear analysis of porous elastically supported FGM plate under various loading Rahul Kumar1*, Achchhe Lal2, B.N. Singh3, and Jeeoot Singh4 Department of Mechanical Engineering

1,2

SVNIT Surat 395007 India Department of Aerospace Engineering

3

Indian Institute of Technology Kharagpur, West Bengal 721302 India Department of Mechanical Engineering

4

Madan Mohan Malaviya University of Technology, Gorakhpur, U.P, 273010, India

*Corresponding Author: [email protected]

Abstract

In this paper, nonlinear transverse central deflection and stress analysis of a functionally graded porous material (FGPM) are investigated by higher order shear deformation theory (HSDT) based on multiquadrics radial basic function (MQ-RBF) meshfree method under various types of loadings and Elastic foundations. The material property of the FGM plate is assumed to vary in the thickness direction and is estimated through the modified Power law homogenization technique with three types of porosity distribution. The governing differential equations (GDEs) are derived from the energy principle containing the porosity effects with the aid of the von Kármán kinematic assumptions and linearized using quadratic extrapolation technique. Convergence and validation studies have been carried out to demonstrate the stability and efficiency of the present method. Numerical results for various types of load, grading index, porosity distribution, porosity index, different span to thickness ratios, and effect of the foundation have been presented. Keywords: FGM plate; Meshfree method; Bending; Porosity volume fraction; Transverse loads; Elastic foundations

1.

Introduction

Continuous improvement in the structural research in industry has imposed the need to upgrade conventional materials with advanced and modern ones wherever possible. FGM is one of the unique heterogeneous materials which was firstly proposed by a group of Japanese scientists in 1984 [1]. FGM is the nonhomogenous materials in which two or more materials (mainly ceramics and metals) combine and the gradation of material properties vary from one phase to other phases by predetermined manner. Due to their unique functional properties such as the large mechanical strength, toughness, and high-temperature resistance with excellent corrosion resistance, it became demanding materials. FGM has been widely used in building structures, pressure vessels, marine ships, automobile industries, and other allied applications. In most of the studies, many investigations have been examined the linear bending analysis of FGM plate with and without porous medium and elastic effects. Qian et al., [2] examined static and dynamic deformation of a square FGM elastic plate by using meshless local PetrovGalerkin (MLPG) method. Ferreira et al., [3] used MQ-RBF based meshfree method for bending analysis of simply supported(SS) FGM plate. Bending analysis of SS FGM plate subjected to a transverse uniformly distributed load (UDL) was examined by Zenkour, [4]. The bending analysis of SS FGM plate resting on an elastic foundation by using proposed

hyperbolic higher order shear deformation theory (HSDT) was carried out by Benyoucef et al., [5]. Mantari et al., [6] proposed new HSDT for the bending analysis of FGM plate and GDEs and boundary conditions are derived by employing the principle of virtual work. Kulkarni et al.,[7] investigated the analytical solution of the FGM plate for bending and buckling analysis. Mantari, [8] investigated bending analysis of FGM plate by using four variables polynomial quasi-3D HSDT. Tran et al., [9] investigated bending, dynamic, and buckling analysis of FGM plate by using HSDT and isogeometric approach. Zenkour, [10] studied bending analysis of porous functionally graded single-layered and sandwich plates.[11] Kim et al., [12] investigated bending, buckling, and, vibration analysis of FGM porous micro-plates with three porosity distributions. Shahsavari et al., [13] investigated vibration analysis of porous FGM plate resting on elastic foundation. Huang et al., investigated exact solution for the bending analysis of FGM plate resting on foundation. Malekzadeh, [14] studied 3-D free vibration analysis of FGM plate resting on elastic foundation by using differential quadrature method. Since a few decades, many numbers of nonlinear bending responses of FGM plate investigations have been done by many researchers. Khabbaz et al., [15] investigated nonlinear bending analysis of FGM plate using classical plate theory(CPT), first-order shear deformation theory(FSDT) and third-order shear deformation (THSDT). on-Karman nonlinearity technique is carried out for investigation. Yang and Shen, [16]investigated the nonlinear bending response of FGM plates subjected to a uniform transverse load with a uniform temperature rise based on Reddy’s HSDT. Naffa and Al-Gahtani, [17] investigated nonlinear response of thin FGM plate by using polynomial radial basis function (RBF) in which no need for shape parameters are required. Zhao and Liew, [18] used the element-free kp-Ritz method to investigated nonlinear analysis of FGM plate by FSDT and von Kármán nonlinearity technique. Alinia and Ghannadpour,[19] investigated the nonlinear analysis of FGM plate by using von Kármán , and material properties vary with the exponential law distribution. Kumar et al., [20] studied nonlinear analysis of FGM plate by using HSDT. The nonlinear equations are obtained by Navier’s method and solved using Newton Raphson iterative method. Singh and Shukla, [21] investigated the nonlinear analysis of FGM plate by using the RBF method. Singha et al., [22] studied the nonlinear response of FGM plate based on FSDT and by using the finite element method, and the nonlinear equations are solved through the Newton–Raphson iteration technique. Zhang, [23]calculated the nonlinear bending response of FGM rectangular plates resting on twoparameter elastic foundations by using Ritz method. Nguyen et al., [24] carried out improved moving Kriging meshfree method for the nonlinear analysis and dynamic analyses of FGM plate. The nonlinearity of plates was solved through a Lagrangian approach based on the von Kármán strain assumptions. Van Do and Lee, [25] carried out a modified mesh-free radial point interpolation method for the nonlinear bending response of FGM plate by using HSDT and new RBF. The nonlinearity of plates was solved through a modified Newton–Raphson iterative technique based on the von Kármán strain assumptions. Shen and Wang, [26] investigated the non linear bending response of FGM plate under thermal effects and resting on an elastic foundation. According to the literature review, most of the researches used uniformly distributed load (UDL) and sinusoidal distributed load for the nonlinear bending of FGM plate using FSDT. Moreover, to the best of author’s knowledge, very little work has been attempted by researchers for analysis of porous FGM plate supported by the elastic foundation with various types of transverse load using RBF based meshfree method. The objectives of the present analysis are to use the MQ-RBF based meshfree method evaluate the nonlinear behaviors of porous FGM plate with different porosity models and supported by an elastic foundations. Also to see the effect of different porosity

distributions, porosity volume index, and gradient index affecting the structural behavior of porous FGM plate Further, the effect of span to thickness ratio, foundation parameters, and boundary conditions will also be investigated. 2.

Mathematical modeling

2.1 Displacement field in FGM plate A porous FGM plate is one of the fascinated of two different material phases, which are metal and ceramic since decades because of its high properties and specification. The typical porous FGM plate with length ‘a’ width ‘b,’ and thickness ‘h’ is taken with the coordinates x, y and z directions. Mid-plane (z=0) of the plate is considered as the reference plane which is shown in Figure 1. The top surface and bottom surface is taken as pure ceramic and pure metal respectively. It is assumed that the grading index ‘n’ of the material phases vary according to thickness.

Figure 1 Geometry and coordinate system of porous FGM plate resting on elastic foundations The displacement field based on the five variables HSDT can be expressed as Kumar et al.[27] w0 ( x, y)  f (z) x ( x, y) x w ( x, y)  f (z) y ( x, y) v  v0 ( x, y)  z 0 y

u  u0 ( x, y)  z

(1)

w  w0  x, y 

The displacement five variables in the x, y, z directions and the rotations about the y- and xaxes are considered as u0 , v0 , w0 ,  x and  y respectively. kw and ks denote the Winkler modulus and the shear modulus of the surrounding elastic medium, respectively

f (z) is

transverse

shear

deformation

function

which

is

expressed

as[27]

3

3p z p   z where p  0.9 4h h

2.2 Strain-displacement equations Strain-displacement equations with von Kármán nonlinear terms are expressed as[19] : 2  u 1  w 2   u0  2 w0  x 1  w0  z f z            x 2  x  x 2   x  x 2  x  2    2 2 v 1  w    v0  z  w0  f  z   y  1  w0   xx          y y 2  y  y 2    y 2  y       yy   y  w0   w0  2 w0   u0 v0    u v  w   w      2z  f z x  f z    xy            x xy y x  x   y    y x  x   y    y   yz   v w   f  z        y  zx   z y   z  u w   f  z      x   z  z x    

                  

(2)

2.3 Constitutive relations of FGM structures The FGM is a nonhomogeneous material with the combination of two materials mainly ceramics and metal.The material properties of the porous FGM plate varying the constituents of materials from one surface to another surface in a predetermined profile. Porous FGM can be defined by the variation in the porosity fraction ‘P’ and grading index ‘n’. In the present analysis, the modified Power law of homogenization techniques with an assessment of three porosity distributions of porous FGM plate is considered along the thickness direction. Hence methods for calculating the effective material properties according to the thickness of the plate are discussed. 2.4 Modified Power law function A typical material property of the FGPM through the plate thickness is assumed to be represented by a power-law which is based on the linear rule of the mixture and has been extensively used to carry out by researchers for the analysis of P-FGPM. For porosity distributions, effective material properties across the plate thickness are given as n    2z  h    E m  1  Xp  ,  h / 2  z  h / 2  E(z)  E c  E m       2h   

(3)

where, ‘E’ is Young’s modulus and the subscripts m and c refers to metal and ceramic, respectively, P is the porosity index (0 
Bottom enhanced porosity distribution, (BPD) [12]

(4)

   z   Xp   P cos    0.5    2 h     

(5)

Top enhanced porosity distribution, (TPD)[12]    z   Xp   P cos    0.5    2 h     

(6)

2.5 Stress-Strain Relations For elastic and FGPM, the 2-D constitutive relations for obtaining stress can be written as[7] xx  Q Q 0 0 0  xx     11 12   0   yy  0  yy  Q12 Q22 0      0 Q66 0 0   xy   xy   0     0 0 Q44 0   yz  0  yz          0 0 0 0 Q 55  zx    zx 



Where

(7)

and  are the stress vector and strain vector respectively. E ( z ) vary through-

thickness direction and ν, is assumed as constant and Qi , j is plane stress-reduced stiffness and is given below. Q1 1  Q 2 2 

E (z)

1    2

,

Q1 2 

E (z)

1    2

, Q 44  Q55  Q 66 

E (z) 2 1  v 

2.6 Governing Differential Equations Energy principle is used herein to derive the governing differential equations of the FGM plate along with variationally admissible boundary conditions. The Potential energy can be expressed as V  U U f W (8)

Where U=strain energy, Uf= strain energy of the elastic foundation and V= work done by the distributed transverse load The strain energy (U) of the FGM plate is expressed by [28] 1 U 2

h 2

  ( 

h 2

   yy  yy   xy  xy   yz  yz   xz  xz ) dz dA

(9)

xx xx

A

Work done by the distributed transverse load is expressed by W 

q

z

w dA

A

(10)

In which qz is the transverse load applied on FGPM plate. The strain energy of the elastic foundation (Uf) can be expressed as: [28]   w  2  w  2   1 2 (10.1)   dA  Uf  kw( w )  ks      x   y    2  A     Where ks and kw are the equivalent shear and Winkler foundation parameters. ks is used to prevent lateral deformation, while kw is used for transverse deformation. This type of model

is the most appropriate for design perspective for structures standing on elastic foundations like rail coaches, truck bodies etc.. The axial force resultants, N ij the bending moment resultants M ij , the additional moment resultants related to the transverse shear function M ijf and the transverse shear force resultants Q xf and Qyf are expressed as:



 h /2

 

N ij , M ij , M ijf 

(i j , zi j , f ( z )i j ) dz

(11)

- h /2

Q

f x

 h /2

  

, Qyf 

xz ,

- h /2

 f ( z )   yz   dz  z 



(12)

The stiffness coefficients of the PFGM plates can be written as: h /2

Aij , Bij , Dij , Eij , Fij , H ij 

 E ( z )   1, z , z 2 , f ( z ), z f ( z ), f 2 ( z )  dz i, j = 1,2,6  2   1 v    h /2 







2   E ( z )   f  z    Aij    dz   2(1  v)   z     h /2   

(13)

h /2



i, j = 4, 5

(14)

By solving the U, V and the Uf and substituting into variational calculus and solving integration by parts, the governing differential equations of the plate are obtained by collecting the coefficients of δu0, δv0, δw0,  x and y can be expressed as: u0 :

v0 :

N xx N xy  0 x y N xy



x

N yy y

0

N xy  N w0 :  xx  y  x  Ny

 x :

 y :

 2 w0 y

2

(15)

(16)

  wo   N yy N xy       x   x   y

 2 N xy

2   2 wo  2 wo  2 M xy  2 w0  2 M xx  M yy       kw w ks 2    0  2 xy xy x 2 y 2 y 2  x

f M xxf M xy   Qxf  0 x y

M xyf x



M yyf y

  wo   2 w0     Nx x 2   y 

 Qyf  0

(17)

   qz 

(18)

(19)

2.7 Boundary conditions The boundary conditions of the FGM plate may take the following boundary conditions and express as [29]: With simply supported boundary condition SS

x  0 , a : N xx  0, v0  0, w0  0, M xx  0,  y  0

(20)

y  0 , b : u 0  0, N yy  0, w0  0,  x  0, M yy  0

Clamped boundary condition CC.

x  0 , a : u 0  0, v0  0, w0  0,  x  0,  y  0

(21)

y  0 , b : u 0  0, v0  0, w0  0,  x  0,  y  0

The governing differential equations in terms of displacements are expressed in the Appendix -A. The governing differential equations along with the appropriate boundary conditions are used to solve using MQ-RBF based meshfree method. 3.

Solution methodology

MQ-RBF (g) is used to discretize the governing differential equations in the space domain and approximate the solution on the principle of interpolation of scattered data over the entire domain. The data point in and over boundary are connected through a special kind of polynomial in terms of the radial distance between nodes which is known as RBF. For the present analysis, MQ-RBF is taken and which is expressed as: Multiquadrics function (MQ-RBF) g   r  c 2

Where, r  X  X j 

x  xj    y  yj  2

2

2



m

2

 a   b ; c  .       nx   n y

  

2

, ‘m’ and ‘c’ are shape parameters. nx and n y are the

number of divisions along with the ‘a’ and ‘b’ respectively. α is a constant that governs the value of ‘c’ for interior and boundary nodes for MQ-RBF. Any variable u can be interpolated in the form of the radial distance between nodes using the following expression [27]:



N

u    uj g X  X j , c j1



(22)

Where, N is a total number of nodes, which is equal to the summation of boundary nodes NB and domain interior nodes NI, uj is unknown coefficient, g  X  X j , c  is radial basis function, X  X j

is the distance between the nodes.

The static problem in terms of RBF can be expressed as [27]: [ K ]L f l  [ F ]  5 N 1   0 L     [ K ]B  5 N 5 N  5 N 1

Where, And

[ K ]L f l  [ K ]L  [ K ]Lf

[ K ]B a

matrix of size 5NBx5N will be obtained using equation (20) or (21).

(23)

3.1 Linearization Technique In the present work, quadratic extrapolation [30] is used for total linearization for nonlinear equations. Nonlinear terms are calculated through a quadratic extrapolation technique at every step of load and transferred to the right hand side as a pseudo load vector, updating the load vector. In this procedure, the linear terms are kept in the left hand side. At the first iteration of every step, nonlinear terms are calculated using the quadratic extrapolation scheme. In subsequent iterations at each step, functions are predicted as the mean of the values at current and preceding iterations. For the first iteration, the predicted value δj is extrapolated quadratically from the values of δ at the three preceding steps as: ( j )  1( j 1)  2 ( j  2)  3( j 3) (24)

Where η1, η2, and η3 are the coefficients of the quadratic extrapolation scheme and take the following values during the initial steps of the load as.[30]. η1=1, η2=0, η3=0; for j=1; η1=2, η2=-1, η3=0; for j=2; η1=3, η2=-3, η3=1; for j ≥3; 3.2 Discretization of Governing Differential Equations The unknown field variables u0 , v0 , w0 ,  x and  y appearing in governing differential equations is assumed in terms of radial basis function as: N









u0 ( x, y )   uj0 g X  X j , c , v0 ( x, y )    vj g  X  X j , c  , w0 ( x, y )  j 1 N

N

0

j 1

 g  X  X N

j 1

w0 j

j

,c



 x ( x, y )    j x g X  X j , c ,  y ( x, y )    j g  X  X j , c  j 1

N

y

j 1

The governing differential equations are discretized and finally expressed in compact matrix form as:

(25.1)

 [ K ]I  [ F ]  5 N 1   L  [ K ]   0 5 N 1  B  5 N 5 N 1

 K   [ F ]     I   L  [ K ]B  0 

(25.2)

The solution methodology for nonlinear flexural analysis is taken as [31]. Where,

  [ u

0



 v0  w 0  x  y ]T 5Nx1

[K]I  [K]L +[K]Lf  [K]NL is 5NIx5N matrix

[K]B is 5NBx5N matrix and is obtained from equations of boundary conditions.

(26)

[ K l1u ]( NI , N )  [ K l 2u ]( NI , N )  [ K ]L  [ K l 3u ]( NI , N )  [ K l 4u ]( NI , N )  [ K l 5u ]( NI , N ) 

[ K l1v ]( NI , N ) [ K l1w ]( NI , N ) [ K l1x ]( NI , N ) [ K l1 y ]( NI , N )   [ K l 2 v ]( NI , N ) [ K l 2 w ]( NI , N ) [ K l 2 x ]( NI , N ) [ K l 2  y ]( NI , N )   [ K l 3v ]( NI , N ) [ K l 3 w ]( NI , N ) [ K l 3x ]( NI , N ) [ K l 3 y ]( NI , N )   [ K l 4 v ]( NI , N ) [ K l 4 w ]( NI , N ) [ K l 4 x ]( NI , N ) [ K l 4  y ]( NI , N )   [ K l 5v ]( NI , N ) [ K l 5 w ]( NI , N ) [ K l 5x ]( NI , N ) [ K l 5 y ]( NI , N )   (5 NI ,5 N )

 [0]( NI , N ) [0]( NI , N ) [0]( NI , N ) [0]( NI , N ) [0]( NI , N )   [0]( NI , N ) [0]( NI , N ) [0]( NI , N ) [0]( NI , N )  [0]( NI , N )   2 2      [ K ]Lf  [0]( NI , N ) [0]( NI , N ) [k w  k s  2  2 ]( NI , N ) [0]( NI , N ) [0]( NI , N )   x y     [0]( NI , N ) [0]( NI , N ) [0]( NI , N ) [ K 4NLw ]( NI , N ) [0]( NI , N )   NL [0]( NI , N ) [ K 5 w ]( NI , N ) [0]( NI , N ) [0]( NI , N )  (5 NI ,5 N ) [0]( NI , N )

[ K ]NL

[0]( NI , N )  [0]( NI , N )   [ K 3NL u ]( NI , N )  [0]( NI , N )  [0]( NI , N )

[0]( NI , N )

[ K1NL w ]( NI , N )

[0]( NI , N )

[0]( NI , N )

[ K 2NLw ]( NI , N )

[0]( NI , N )

NL NL [ K 3NL v ]( NI , N ) [ K 3 w ]( NI , N ) [ K 3 x ]( NI , N )

[0]( NI , N )

[ K 4NLw ]( NI , N )

[0]( NI , N )

[0]( NI , N )

NL 5 w ( NI , N )

[0]( NI , N )

[K

]

F L  0( NI ,1) 0( NI ,1) qz ( NI ,1) 0( NI ,1) 0( NI ,1) 

4.

  [0]( NI , N )   [ K 3NL  y ]( NI , N )   [0]( NI , N )   [0]( NI , N )  (5 NI ,5 N ) [0]( NI , N )

(27)

(28)

(29)

T

(30)

Results and discussion

This section deals with the nonlinear bending analysis of porous FGM plates resting on two parameters Pasternak elastic foundation. Uniformly distributed load (UDL), parabolic load (PL), uniformly varying load (UVL), uniformly distributed central patch load (UCPL) and uniformly distributed symmetry corner patch load (USCPL) are considered in the present analysis and shown in Table 1 (a)-(e). In UCPL, the load is applied at 5X5 nodes patch in the center of the plate. So, the load concentrated in 25 nodes out of 169 nodes. In USCPL, the load is applied at 4X4 nodes patch in all the corners of the plate in which load concentrated in 64 nodes out of 169 nodes. The materials properties of the FGM are listed in Table 2 for the computation of numerical results, unless specified otherwise. The dimensionless maximum deflection, central deflection, stresses, and the load parameters are expressed as:

12(1  2 )q0a 4 h  xx   h xy ,   h xz wc q  ,   , xy xz w , z xx q 0a q 0a q 0a Ec h 4 h

Where qo is input load parameter. The parameters  xx  xy and  xz are dimensionless stresses along the longitudinal direction and shear direction of x-y and x-z direction respectively. The dimensionless Winkler-Pasternak elastic foundation parameters are presented as:

12a 4 kw(1   2 ) 12a 2 ks(1   2 ) Kw  , Ks  Emh3 Emh3 Where Kw and Ks are the input foundation parameters. Table 1 Different types of transverse loads namely acting on porous FGM plate Sr.No Type of transverse load (a) Uniformly distributed load UDL : qz 2

(b)

Parabolic load

(c)

Uniformly varying load

(d)

Uniformly distributed central patch load

(e)

Uniformly distributed symmetry corner patch load

Table 2 Mechanical properties of FGM Types of Functionally graded material FGM-1 FGM-2

Metal (Al) Ceramic (Al2O3) Metal (Al) Ceramic (ZrO2)

 y PL : qz   b  y UVL : qz   b 169 UCPL : qz 25 169 USCPL : qz 64

Properties E (GPa) 70 380 70 151

 (kg/m3)



2702 3800 2702 3000

0.3 0.3 0.3 0.3

5.1 Convergence and validation In order to demonstrate the accuracy and stability of the present methodology, convergence and validation study for the non-linear bending analysis of FGM are carried out for SS and CC boundary condition in Figure 2(a)-(b). It can be seen that the present results are in good agreement with Singha et al., [22] and converged after 13x13 nodes with convergence rate less than 0.6% for both SS and CC.

1.4 1.2 1.0

wc

0.8 0.6

7X7 9X9 11X11 13X13 15X15 Ref.

0.4 0.2 0.0 0

50

100

150

200

250

300

350

400

450

500

Load

(a) 0.8 0.7 0.6 0.5

w c 0.4 0.3

7X7 9X9 11X11 13X13 15X15 Ref.

0.2 0.1 0.0 0

50

100

150

200

250

300

350

400

450

500

Load

(b)

Figure 2 Validation and convergence of the FGM-1 plate under UDL (n=0.5, a/h=100, load step 25); (a) convergence and validation under SS boundary condition and (b) convergence and validation under CC boundary condition. Figure 3 (a)-(b) shows the convergence study of square SS FGM-1 plate under PL and UVL for n’=0.5, load step 25 and a/h=100. It can be observed that for both types of loads transverse deflection is converged within 0.6 % after 13x13 nodes. So, 15x15 nodes are taken for the further computation of numerical results.

0.7 0.6 0.5

wc

0.4 0.3

7X7 9X9 11X11 13X13 15X15

0.2 0.1 0.0 0

50

100

150

200

250

300

350

400

450

500

Load

(a)

1.0 0.9 0.8 0.7 0.6

w c 0.5 0.4

7X7 9X9 11X11 13X13 15X15

0.3 0.2 0.1 0.0 0

50

100

150

200

250

300

350

400

450

500

Load

(b)

Figure 3 Convergence study of normalized central non- linear deflection of SS FGM-1 plate; (a) parabolic load (b) Uniformly varying load 5.2 Parametric Studies of transverse deflection and stresses Detailed parametric studies are carried out to investigate the nonlinear bending analysis of porous FGM-2 plate under five types of transverse load such as UDL, PL, UVL, UCPL, and USCPL. The parametric studies are focused on the evaluation of transverse deflection and stress with the variation of various transverse loads, grading index, porosity distribution, porosity index, span to thickness ratio, and foundation parameters.

Table 3 represents the effects of various types of loads on the normalized nonlinear deflection of FGM-2 plate for a/h=10, Kw=100, and Ks=100, load step=25 and n=1 with SS and CC boundary conditions. It can be seen from UCPL predicts highest normalized central deflection followed by UDL, while least for PL for both SS and CC boundary conditions. It is because of change in normalized central deflection is that in all loading condition the magnitude of the transverse load is the same but the loading areas are changed. The magnitudes of the normalized nonlinear deflection in the SS plates are larger as compared to the clamped plates because clamped plates are stiffer than the SS plates. Table 4 Nonlinear central deflection of FGM-2 plate under various types of loads Load (SS) Load (CC) Types of load 100 200 300 400 100 200 300 UDL 0.127 0.247 0.3601 0.4644 0.0902 0.1782 0.2647 PL 0.0371 0.0736 0.1099 0.1454 0.0263 0.0522 0.0784 UVL 0.064 0.1263 0.1876 0.2465 0.0452 0.0898 0.1346 UCPL 0.312 0.5685 0.771 0.9331 0.2166 0.4135 0.5877 USCPL 0.0567 0.111 0.1633 0.2123 0.0414 0.0821 0.1227

400 0.3472 0.1042 0.1784 0.7392 0.1622

Error! Not a valid bookmark self-reference.4(a)-(b) represents the effect of different types of load on normalized central deflection verses length of plate under SS and CC boundary condition for ‘n’=1, a/h=20, load step=25, Kw=100, Ks=100. It can be seen from UCPL predicts highest normalized central deflection followed by UDL and is least predicted for PL for both SS and CC boundary conditions. 0.0

Normalized central deflection

-0.1

-0.2

-0.3

-0.4

UDL PL

-0.5

UVL UCPL

-0.6

USCPL 0.0

0.2

0.4

0.6

Length of plate

(a)

0.8

1.0

Normalized central deflection

0.0

-0.1

-0.2

-0.3

UDL PL

-0.4

UVL UCPL USCPL

-0.5 0.0

0.2

0.4

0.6

0.8

1.0

Length of plate

(b)

Figure 4 Nonlinear central deflections of FGM-2 plate under various types of the load along the central line for (a) SS boundary condition and (b) CC boundary condition. Figure 5(a)-(b) shows the influences of foundation parameters on normalized central nonlinear deflection of square SS and CC FGM-2 plate under USCPL for ‘n’= 0.5, a/h= 10 and load step=25 . At load=200, the effect of Kw=0, Ks=50 to FGM plate without foundation is 51% and the effect of Kw=50,Ks=0 to FGM plate without foundation is 9%. So, the effect of Ks is more dominating than Kw and increasing the value of Kw and Ks, the normalized central deflection decrease with respect to the FGM plate without foundation effect in both the boundary conditions. It can be noticed that by increasing the values of Ks and Kw, the normalized central nonlinear deflection of FGM-2 plate is suddenly weaker by increasing Ks as compare to Kw.

0.8 Kw=0,Ks=0 Kw=50,Ks=0 Kw=0,Ks=50 Kw=0,Ks=100 Kw=100,Ks=0 Kw=200,Ks=0 Kw=0,Ks=200 Kw=0,Ks=400 Kw=400,Ks=0

Normalized central deflection

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

100

200

300

400

500

400

500

Load

(a)

Normalized central deflection

0.5 Kw=0,Ks=0 Kw=0,Ks=50 Kw=50,Ks=0 Kw=0,Ks=100 Kw=100,Ks=0 Kw=0,Ks=200 Kw=200,Ks=0 Kw=400,Ks=0 Kw=0,Ks=400

0.4

0.3

0.2

0.1

0.0 0

100

200

300

Load

(b)

0.0

Normalized central deflection

-0.2 -0.4 -0.6 -0.8 -1.0 -1.2

Ks=0, Kw=0 ,

Ks=0, Kw=50

Ks=0, Kw=100 ,

-1.4

Ks=50, Kw=0 ,

Ks=0, Kw=400 Ks=100, Kw=0

Ks=400, Kw=0

-1.6 0.0

0.2

0.4

0.6

0.8

1.0

Length of plate

Figure 5 (a)

Normalized central deflection

0.0

-0.2

-0.4

-0.6

Ks=0, Kw=0 ,

-0.8

Ks=0, Kw=50

Ks=0, Kw=100 , Ks=50, Kw=0 ,

Ks=0, Kw=400 Ks=100, Kw=0

Ks=400, Kw=0

-1.0 0.0

0.2

0.4

0.6

0.8

1.0

Length of plate

(b) Figure Figure 6 (a)-(b) represents the effect of foundation parameters on the normalized central nonlinear deflection of FGM plate along the central line under SS and CC for a/h=20, n=1, load 200, and UCPL. It can be seen that the effect of Ks is more dominating as compared to Kw. And by increasing the value of Kw and Ks, the normalized central deflection decreases.

0.0

Normalized central deflection

-0.2 -0.4 -0.6 -0.8 -1.0 -1.2

Ks=0, Kw=0 ,

Ks=0, Kw=50

Ks=0, Kw=100 ,

-1.4

Ks=0, Kw=400

Ks=50, Kw=0 ,

Ks=100, Kw=0

Ks=400, Kw=0

-1.6 0.0

0.2

0.4

0.6

0.8

1.0

0.8

1.0

Length of plate

(a)

Normalized central deflection

0.0

-0.2

-0.4

-0.6

Ks=0, Kw=0 ,

-0.8

Ks=0, Kw=50

Ks=0, Kw=100 ,

Ks=0, Kw=400

Ks=50, Kw=0 ,

Ks=100, Kw=0

Ks=400, Kw=0

-1.0 0.0

0.2

0.4

0.6

Length of plate

(b)

0.0

Normalized central deflection

-0.2 -0.4 -0.6 -0.8 -1.0 -1.2

Ks=0, Kw=0 ,

Ks=0, Kw=50

Ks=0, Kw=100 ,

-1.4

Ks=50, Kw=0 ,

Ks=0, Kw=400 Ks=100, Kw=0

Ks=400, Kw=0

-1.6 0.0

0.2

0.4

0.6

0.8

1.0

Length of plate

Figure 6 (a)

Normalized central deflection

0.0

-0.2

-0.4

-0.6

Ks=0, Kw=0 ,

-0.8

Ks=0, Kw=50

Ks=0, Kw=100 , Ks=50, Kw=0 ,

Ks=0, Kw=400 Ks=100, Kw=0

Ks=400, Kw=0

-1.0 0.0

0.2

0.4

0.6

0.8

1.0

Length of plate

(b) Figure Figure 7 represents the effects of three types of porosity distribution with various porosity indexes on the normalized central deflection of SS FGM plate for n=1, a/h=10, load step 25, UDL, Kw=100, Ks=100.The first model is SCPD in with high density of porosity at the center and the symmetric with respect to the midplane of FGM Plate. The second model is BPD in which the porosity density higher at the bottom and the third model is TPD in with porosity density higher at the top. It can be observed that TDP predicts the highest porosity distribution effect on normalized central deflection followed by SCPD and least is predicted by BPD. The reason is that the high density of porosity at ceramic makes the plate softer and lighter as compared to other two porosity distribution. The SCPD predicts lower normalized

central deflection that TDP and BPD because the plate with SCPD is stiffer due to porosity denser at the center than TDP and BPD and ceramic and a metal layer, are more sensitive than the midplane surface. By increasing the porosity index in all the porosity distributions, the normalized central deflection increase because by increasing the value of P, the porosity is denser and due to which the plate start lighter and softer. So, the porosity distribution location is a critical parameter for predicting the normalized central deflections.

Normalized centeral deflection

1.2

1.0

0.8

Pure FGM SCPD,P=0.1 SCPD,P=0.2 SCPD,P=0.3 BPD,P=0.1 BPD,P=0.2 BPD,P=0.3 TPD,P=0.1 TPD,P=0.2 TPD,P=0.3

0.6

0.4

0.2

0.0 0

100

200

300

400

500

Load

Figure 7 Effect of porosity distribution on normalized central nonlinear deflections of SS porous FGM plate. Figure 8 represents the influence of gradient index (n) on normalized central deflections of SS FGM-2 plates is presented with five different types of load on pure FGM-2 plate for a/h=10, load step 25, Load =150, Kw=0, Ks=0. It can be seen that the increase in the value of grading index increases the normalized central deflection, and all the loads predict the same nature and the effect of n is decreased after n=5. The increase in ‘n’ transforms the pure ceramic into a combination of ceramic and metals phases. The combination increases the stiffness of the pure FGM plate without foundation effect.

Normalized central deflection

1.0 UDL PL UVL UCPL USCPL

0.8

0.6

0.4

0.2

0

2

4

6

8

10

Grading index

Figure 8 Effect of grading index on normalized central nonlinear deflections of SS pure FGM-2 plate. Figure 9 shows the influence of gradient index (n) on normalized central deflections of SS porous FGM-2 plates under UCPL load for a/h=20, SCPD, P=0.2, load step 25, PL, Kw=100, Ks=100. It can be seen that by increasing the value of n, the normalized central nonlinear deflections increase and after ‘n’=5, the effect is negligible.

Normalized central deflection

0.20

0.15

0.10

n=0 n=0.5 n=1 n=2 n=5 n=10

0.05

0.00 0

100

200

300

400

500

Load

Figure 9 Effect of grading index on normalized central nonlinear deflections of SS FGM-2 plate.

Figure 20 represents the effect of span to thickness ratio of porous FGM-2 plate for n=1, P=0.2, load step 25, UVL, Kw=50, Ks=50. It can be seen that the normalized central deflection decrease from thick to a thin plate and after a/h=20, the effect is negligible.

Normalized centeral deflection

0.5

0.4

0.3

0.2

a/h=5 a/h=10 a/h=20 a/h=50 a/h=100

0.1

0.0 0

100

200

300

400

500

Load

Figure 20 Effect of span to thickness ratio on normalized central nonlinear deflections of SS FGM-2 plate resting on elastic foundation.

0.5 0.4 0.3 0.2 0.1

z/h

UDL PL UVL UCPL USCPL

0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -7

-6

-5

-4

-3

-2

-1

0

a b   xx  , , z  2 2 

(a)

1

2

3

4

5

0.5 0.4 0.3 0.2 0.1

z/h

0.0 -0.1

UDL PL UVL UCPL USCPL

-0.2 -0.3 -0.4 -0.5 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

 xy  0,0, z 

1.5

2.0

2.5

(b)

0.5

UDL PL UVL UCPL USCPL

0.4 0.3 0.2 0.1

z/h

0.0 -0.1 -0.2 -0.3 -0.4 -0.5 0.000

0.005

0.010

0.015

 xz

0.020

0.025

0.030

0.035

0.040

 b   0, 2 , z   

(c)

Figure 31 Effect of various types of load on normalized stress along with the thickness of the SS porousFGM-2 plate. (a)  xx  a , b , z  ,(b)  xy  0,0, z  ,(c)  xz  0, b , z  2 2





2



Figure 31(a)-(c) represents the effect of various types of load on normalized stresses along with the thickness of SS porous FGM-2 plate for n=1, a/h =20, load 200, SCPD, P=0.2. It can

be seen that  xx for USCPL predicts high-stress value as compared to other loads. The maximum compressive stresses occur on the ceramic layer, and the maximum tensile stresses occur, on the metal layer of the FGM plates, and it is also noticed that maximum tensile stresses are greater than maximum compressive stresses.  xy for the entire load followed the same pattern and predicts maximum at the metal layer. The stress  xz at top and bottom satisfied zero condition and follow parabolic in nature. The maximum stress of UCPL predicts the highest value as compared to other loads and maximum stresses for all the loads are towards the metal layer from midplane. 5.

Conclusions

This study attempted to explore nonlinear responses of porous FGM plate resting on WinklerPasternak elastic foundation with various transverse loadings. The plate formulation is based on five variables algebraic HSDT with von Kármán nonlinearity combined with MQ-RBF meshfree method. The following conclusions are noted from the present study: Among the different types of transverse loading, UCPL shows the highest deflection while PL shows the lowest deflection at the centre. Plate supported by Winkler-Pasternak foundation decreases the deflection of the plate due to increment in the overall stiffness of the plate. It is also observed that shear foundation is more dominant over Winkler foundation due to the distribution over both of the axis. Among the given three types of porosity distribution considered, TDP predicts the highest deflection as compared to BDP and SCPD. With the increase of the porosity index, transverse deflection increases due to lowering strength and stiffness. As the grading index increase, the transverse deflection increases due to the more amount of metal portion is added. The deflection of the plate increases with the decrement in thickness ratio.

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Appendix -A u0 :

N xx N xy  0 x y

  2 u0  2 u0    2 v0  2 v0    3 w0  3 w0  3 w0   A11 A A A B B 2 B            66 12 66 11 12 66 xy xy   x 2 y 2   x3 xy 2 xy 2      2 y 2 y  2x 2x    E12  E E        E11 66 66 xy xy  x 2 y 2       2 2 2    w0  w0  w0   w0  2 w0  w0 A  A A A    0     66 12 66   11 x 2 xy xy  y y 2  x   

v0 :

N xy x



N yy y

             

A-2

f M xxf M xy   Qxf  0  x y

  2u  2 u0    2 v0  2 v0    3 w0  3 w0  3 w0    E11 20  E66 2 E E F F F            12 66 11 12 66 xy xy   x y 2   x3 xy 2 xy 2         2 y 2 y 2 y   2x 2x  2 w0  2 w0  H12    E11 H A H H       E      H11 66 55 22 66 x 66   xy xy   x 2 y 2 y 2 x 2 y 2          2 w0  2 w0  w0  E66 0     E12 xy xy  y  

 y :

A-1

0

  2 u0  2 u0   2v 2v   3 w 3 w  3 w0   A66  A12   A66 20  A22 20    2 B66 2 0  B12 2 0  B22  xy xy   x y   x y x y y 3      2 y 2 y  2x      E22   E66    ( E12  E66 ) 2 2  xy    x  y      2 2 2   w0  w0   w0  2 w0  w0   w0 0   A A A        A66 22 66 12 yx  x  xy y 2  y  x 2     x :

           

M xyf x



M yyf y

     w0     x     

A-3

 Qyf  0 

   2 u0  2 u0    2 v0  2 v0    E66   E12   E66  E22   2 2 xy xy   x y       3 3 3 2 2    w0  w0  w0    x  x    F12  F22  H12    H 66     2 F66 xy xy   yx 2 yx 2 y 3         2 y 2 y  2 w0  2 w0  w0       H 66 x 2  H 22 y 2  A44  y    E12 yx  E66 xy  x          2 2      E  w0  E  w0  w0  0  22 2    66 x 2   y y    

A-4

2 N xy   wo   N yy N xy   wo   N b  w0 w0 :  xx        (Nx  Nx ) 2 y   x   y x   y  x  x

 ( N y  N yb )

 2 w0 y

 2( N xy  N xy b )

2

2  2 M xy   2 w  2 wo  2 w0  2 M xx  M yy   2  k w w0  k s  2o  2 2  x xy xy x y y 2 

   qz 

   3 u0  3 u0  B11 3   B12  2 B66  x xy 2 

   3v0  3v0     B22 3   B12  2 B66  2  y x y   

  4 w0 4 w  4 w0    3  3 x    D11   2 D12  4 D66  2 0 2  D22   F11 3x   F12  2 F66    4 4  x x y y   x xy 2     3 y  3 y    2 u0  2 u0  2 v0  2 v0   w0   2 F F A A   F22     A A        12 66 11 66 12 66  xy xy   x  y 3 x 2 y   x 2 y 2    3 w0  3 w0  3 w0 2 B B    B11   12 66  x3 xy 2 xy 2    2 w0  2 w0    A11  A66 2   x y 2 

2 y  2  y   w0    w0   2x 2x  E E        E11 2  E66  12 66 xy xy   x  x y 2   x   

 w0   2 w0   ( A12  A66 )   xy  x 

 w0   y

A 5

  w0     x  

  2 u0 2v  2 v   w   3 w 3 w  3 w0   w0    ( A12  A66 )  A66 20  A22 20   0    2 B66 2 0  B12 2 0  B22      xy x y   y   x y x y y 3   y    2 y  2  y   w0    2x 2x  2 w0  w0   2 w0  2 w0  w0   w0   E66   E12  E66 E     ( A12  A66 )   A66  A22      22 2 2 2     xy xy xy  x  x y   y    x y 2  y   y   2 2  3 3   2 w0       2    3 w0  3 w0  w0     B  w0   B  2 B   w0  w0  B   w0   2 B B B    B11         12 66 11  12 66 22  2  2   22 y 3   x3 xy 2  x x 2 y  y  x      y      2 w0     2 B12  2 B66   xy 

  2 w0  2 w0  2 w0  2 B66  y 2 x 2  xy

2 2   1  w  1  w     A11  0   A12  0    E11 x  2  x  2  y   x 

  u0  A    11 x  

v0    2 w0  2 w0   A B B        12 11 12  y   x 2 y 2  

 y    2 w0   u0      E12 y   2    A12 x x      

   2 w0   2    x

v0    2 w0  2 w0    B22    A22 y     B12 2 x y 2    

2 2   y    2 w0      u0   v 1  w  1  w     A12  0   A22  0    E12 x    E22   A66 0   2    A66   2  x  2  y   x   y   y y   x      y    2 w0   2 w  2 wo  w w       A66 0 0   E66 x    E66  kw  w0   ks  2o    qz   2  x  x y  y   x   xy x 2   

 2 w0   2 B    66   xy  

   2 w0   2    xy

   2 w0   2    y

CONFLICT OF INTEREST STATEMENT The authors declare that there is no conflict of interest