Nonlinear dynamic analysis of sandwich S-FGM plate resting on pasternak foundation under thermal environment

European Journal of Mechanics / A Solids 76 (2019) 155–179

Contents lists available at ScienceDirect

European Journal of Mechanics / A Solids journal homepage: www.elsevier.com/locate/ejmsol

Nonlinear dynamic analysis of sandwich S-FGM plate resting on pasternak foundation under thermal environment

T

S.J. Singh∗, S.P. Harsha Vibration and Noise Control Laboratory, Mechanical and Industrial Engineering Department, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India

ARTICLE INFO Keywords: Sandwich plate Sigmoid function Thermal environment Functionally graded material Von-karman strains Poincaré maps Airy's function Galerkin approach Nonlinear dynamics

In the present study, stress-function Galerkin (SFG) method is employed to investigate the dynamic characteristics of a new sigmoid law based sandwich functionally graded plate (S-FGM) plates resting of Pasternak elastic foundation in the thermal environment. For modified sigmoid law, a new temperature profile is derived considering 1D steady state heat conduction equation. The Hamiltonian formulation is done to derive governing equations and nonlinearity, due to Von- Karman strains, is worked out using Airy's function in conjunction with Galerkin's method. The time and frequency domain analysis is then performed using a numerical integration scheme and harmonic balance method, respectively. The nonlinear rise in temperature is considered across the thickness due to the temperature difference between the top and the bottom surface of the simply supported plate with immovable edges. Wide-Ranging parametric studies for, linear and nonlinear, frequency and time domain analysis have been performed by taking into consideration the effect of thickness ratio, inhomogeneity parameter, thermal load, and foundation parameter for various configurations of the sandwich plates. Poincaré maps, phase-plane plots and time responses are demonstrated to study the nonlinear dynamics behavior of sandwich S-FGM plate under harmonic excitation. The variation of aspect ratios shows the route to chaos. With the Winkler foundation, the response is chaotic but becomes weakly chaotic with the introduction of the Pasternak type foundation. The dynamic response clearly shows the route to chaos with the varying thermal load from ΔT = 0–600 K. It is observed that the periodicity of the plate behavior is primarily affected by considering different configurations of the sandwich S-FGM plate. The computed results and observations can be utilized as a validation study for future examination for sandwich S-FGM plates.

1. Introduction “Functionally Graded Materials” (FGM) are new advanced material having material properties that varies smoothly and continuously in an utmost required direction. FGMs are introduced in engineering applications working under severe operating conditions and have considerable applications in structural components subjected to extreme temperature. With the advancement of graded material and the requirement of properties like low weight and high strength, have led to the development of functionally graded sandwich plates. The aboveexpressed necessities induce high stiffness to the material which is primarily required for the structures working under huge excitation and harsh conditions. The conventional sandwich plates under severe conditions and with passage of time, results in lessening of the sandwich effect due to delamination of the face sheets. These disadvantages of overlaid composite must be limited by gradually varying the volume fraction of the constituent materials by considering the functionally

∗

graded face sheets, as material properties vary continuously in functionally graded materials. This will eliminate interface problems of composite materials between face sheets and the core, and thus resulting in smooth stress distributions relative to the conventional sandwich plate. Because of these limitations and motivated by the possibility of designing the components with superior physical properties, the FGM concept has triggered a considerable amount of worldwide research activities (Mortensen and Suresh, 1995; Xu et al., 2018). Currently, the static and dynamic analysis of sandwich FGM plates has attracted a lot of researcher due to their inherent property of light weight and high stiffness. The linear and nonlinear studies on sandwich FGM plates have been done by considering variation in material properties according to different homogenization laws. In most of the studies on the sandwich plate with material properties vary according to a simple power law (P-FGM), which has been reported in the literature for linear Eigenvalue problems. In this

Corresponding author. E-mail addresses: [email protected] (S.J. Singh), [email protected] (S.P. Harsha).

https://doi.org/10.1016/j.euromechsol.2019.04.005 Received 5 February 2019; Received in revised form 28 March 2019; Accepted 3 April 2019 Available online 11 April 2019 0997-7538/ © 2019 Elsevier Masson SAS. All rights reserved.

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

framework, Li et al. (2008) employed 3D exact solution to study free vibration of symmetric and unsymmetric sandwich plates for different configurations viz. homogenous core and FGM facesheets, FGM core and homogenous facesheets. The solution was obtained by combining the Ritz method and Chebyshev polynomial for clamped and simply supported sandwich plates. On the same track, Meiche et al. and Thai et al. presented a new hyperbolic (El Meiche et al., 2011) and first-order (Thai et al., 2014) shear deformation theory for free vibration and buckling analysis of symmetric and unsymmetric sandwich plates. The sandwich plates with homogenous core and FGM face sheets were analyzed for various thickness ratio and volume fraction exponent. Zenkour et al. (Zenkour, 2005) performed the buckling and free vibration analysis of a symmetric sandwich plate with homogenous core and FGM face sheets. Nguyen et al. (2014) proposed a new inverse trigonometric shear deformation theory for the static, buckling and free vibration analyses of single layered and FGM sandwich plates. The solution was obtained for three types of sandwich FGM plates: single layered FGM plate, sandwich plates with FGM core and sandwich plates with FGM faces. The exact solution is obtained using Navier's method for a sandwich plate with all edges simply supported (El Meiche et al., 2011; Zenkour, 2005; Nguyen et al., 2014). Studies on P-FGM sandwich plate with thickness stretching effect have been done by many researchers, since avoiding thickness stretching effect will give erroneous results in thick plates. But, there is not such a significant effect on moderately thick and thin plates, when considering the same (Neves et al., 2013a). In this context, Neves et al. (2013a) and Natarajan et al. (Natarajan and Manickam, 2012) derived quasi 3D higher order shear deformation theory for bending and free vibration, analysis of single layered and sandwich FGM plates. Three different types of symmetric and unsymmetric sandwich plate configurations are analyzed, specifically: single layered FGM plate, FGM core with homogenous face sheets and, a homogenous core with FGM face sheets. The solution was obtained using meshless technique (Neves et al., 2013a) in conjunction with Radial Basis Function (RBF) to handle boundary conditions and QUAD-8 shear flexible element (Natarajan and Manickam, 2012). Bessaim et al. (2013) developed new higher order and normal shear deformation theory for static and free vibration analysis of symmetric and unsymmetric sandwich plates. Bennoun et al. (2016) developed a new five-variable refined plate theory for free vibration analysis of symmetric and unsymmetric sandwich FGM plate. Two common types of sandwich plates were considered, namely, FGM facesheet with hard/soft homogeneous core and the homogeneous facesheet with FGM core. The closed form solution was obtained using Navier's method (Bessaim et al., 2013; Bennoun et al., 2016). Similarly, a lot of research on P-FGM sandwich plate had been reported by researchers for the development of efficient displacement model (Tounsi et al., 2013; Bourada et al., 2012; Merdaci et al., 2011; Abdelaziz et al., 2011) and/or method of obtaining an accurate and precise solution (Xiang et al., 2013). The improvement in composite materials led to the development of more efficient micro-mechanical models, which defines the variation of material property in the direction of thickness and from this perspective, exponential (E-FGM) based variation of material properties was taken into consideration. The study on sandwich E-FGM plate is very scarce and till now only two studies have been reported in the literature. Sobhy et al. (Sobhy, 2013) and Meziane et al. (2014) investigated free vibrations and critical buckling loads for various types of E-FGM sandwich plates. New exponential law distribution in terms of the volume fractions of the constituents was proposed (Sobhy, 2013) and further implemented by Meziane et al. (2014). The analysis was performed for a symmetric and unsymmetric sandwich plate with FGM face sheets and homogenous core. In addition, Meziane et al. (2014) proposed an efficient and simple refined shear deformation theory that made the novel theory much more pliable to the application. The exact solution for the sandwich plate was obtained using Navier's method and different boundary conditions were implemented by considering Eigen

functions satisfying the geometric boundary conditions. Apart from the above-mentioned homogenization technique, sigmoid law is another micro-mechanical model in which analysis is still lacking and specifically, very little research has been carried out on a single-layered S-FGM plate (Han et al., 2015; Fazzolari, 2016; Lee et al., 2015; Beldjelili et al., 2016; Jung et al., 2014, 2016). The advantages of S-FGM plate over P-FGM plate lies in the fact that the former provides more strength and stiffness under the same working conditions (Duc and Cong, 2015; Nguyen, 2018; Duc et al., 2015; Singh and Harsha, 2019). Recently, a new endeavor has been carried out by Singh et al. (Singh and Harsha, 2019) to study free vibration and buckling analysis of symmetric and unsymmetric sandwich S-FGM plate resting on elastic foundation with different boundary conditions. In this study new modified sigmoid functions have been developed for sandwich plate keeping in mind the smooth variation of constituent material along the thickness of the plate. Three different configurations of sandwich plates were analyzed namely, Isotropic S-FGM, S-FGM facesheets with a homogenous hard/soft core, and homogenous facesheets with FGM core. Navier's method was employed to obtain close form solution and boundary conditions were implemented by considering the shape functions satisfying the geometric boundary conditions. Further, Singh et al. (Singh and Harsha, 2018) performed nonlinear dynamic analysis of sandwich S-FGM plate with FGM core and homogenous facesheets. The solution was obtained using Navier's method in conjunction with Airy's stress function to deal with nonlinearity. Moreover, studies on single layered FGM plates in the thermal environment is also available in the literature, however, studies on sandwich FGM plate in similar environment are few in numbers. In this context, Natarajan et al. (Natarajan and Manickam, 2012) investigated the bending and the free flexural vibration behavior of sandwich FGM plates. They considered two different types of sandwich plates, viz., homogeneous face sheets with FGM core and FGM face sheets with homogeneous hard core. The material was assumed to be graded only in the thickness direction according to a power-law distribution (P-FGM). Zenkour et al. (Zenkour and Alghamdi, 2008, 2010) performed a thermoelastic bending analysis of functionally graded ceramic-metal symmetric sandwich plates. The material properties are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. The core layer was homogeneous and made of an isotropic ceramic material. Tounsi et al. (2013) presented a thermoelastic bending analysis of functionally graded sandwich plates with homogeneous facesheet and FGM core. Material properties of the FGM core were assumed to vary according to a power law distribution (P-FGM) in terms of the volume fractions of the constituents. Pandey et al. (Pandey and Pradyumna, 2015) presented free vibration analysis of two types of FGM sandwich plates, one with homogenous facesheets and functionally graded core and the second with functionally graded facesheets and homogenous core, with nonlinear temperature variation along the thickness. The material properties of both types of FGM sandwich plates are varied according to Mori-Tanaka (MT) scheme and the rule of mixture (ROM). In the light of above discussion, it can be inferred that the available research on the thermo-mechanical analysis of sandwich FGM plates is very limited. Although study on chaotic behavior (Hao et al., 2008; Alijani et al., 2011; Zhang et al., 2010) of single layered P-FGM plate has been done in thermal environment by several researchers. Also, the analysis on single layered S-FGM (Han et al., 2015; Fazzolari, 2016; Lee et al., 2015; Beldjelili et al., 2016; Jung et al., 2014, 2016) plate in the non-thermal environment is also available in the literature. In most of the studies on sandwich plates, research is confined to frequency response analysis only, but research on time response analysis, chaotic vibration, and route to chaos under thermal environment is not available in the literature. This paraphernalia is really important and critical for safe and reliable prediction and good design. In addition, research on nonlinear dynamic analysis under thermal environment considers the effect of temperature only on the structural stiffness of the plate but 156

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

the effect on initial bending of the plate was not considered earlier. Thus, in the present study, nonlinear thermo-mechanical dynamic characteristics of S-FGM sandwich plates resting on Pasternak elastic foundation with a simultaneous combination of non-polynomial shear deformation theory (Joshan et al., 2017) (which was earlier formulated for laminated plates) and stress function has been investigated. The material properties of the constituent materials in terms of volume fraction for the sandwich plate are assumed to vary in the thickness direction according to a new sigmoidal law distribution. In addition, the new temperature profile is derived considering 1D steady state heat conduction equation which has not been derived earlier for sandwich and S-FGM plate in the literature so far as per author's knowledge. Wide-Ranging parametric studies for, linear and nonlinear, frequency and time domain analysis have been performed by taking into consideration the effect of thickness ratio, inhomogeneity parameter, thermal load, and foundation parameter for various configurations of the sandwich plates. These studies are performed for plate resting on Pasternak elastic foundation and subjected to thermo-mechanical loading. The thermo-mechanical analysis of sandwich S-FGM plate is not existing in the available literature to the best of the author's knowledge. Consequently, the computed results and observations can be utilized as a validation study for future examination for sandwich SFGM plates.

in

1 0

2.where,

i

k + k+ 1

=

2

0 for k = 1 for 2 for

,

0 1 2

1

2,

Vt =

3

Ceramic Metal

3. Mathematical formulation In the present formulation, S-FGM plate considered is resting on two-parameter based elastic foundation as shown in Fig. 3. The plate is idealized as a perfect plate (a × b) without any initial geometric deformation and micro-voids that may occur during the fabrication process. 3.1. Displacement field and strains The displacement field is expressed using non-polynomial based inverse hyperbolic shear deformation theory which takes into account the nonlinear variation of transverse shear stresses and strains in order to capture realistic behavior of the plate. Thus, in accordance with the aforementioned variation, the in-plane (u, v) and out-of-plane displacement (w) is given in Eq. (2):

2. Modified sigmoid function and homogenization The existing sigmoid function is used and modified to obtain a volume fraction of constituent materials for sandwich S-FGM plate. The modified functions are obtained keeping in mind, the most vital inherent characteristic of FGM i.e. continuous variation of material properties in terms of volume fraction. The deterioration in material properties is observed with the rise in temperature and it is assumed that the bottom face of the plate is always at room temperature (300 K) as shown in Fig. 1. The following type of single layered and sandwich FGM plates are considered in the present paper:

u x , y, z, t = u¯ x , y , t

z

( ) + f(z )

x

x , y, t

v x , y , z , t = v¯ x , y , t

z

( ) + f(z )

y

x, y, t

w¯ x

w¯ y

(2)

w (x , y , z , t ) = w¯ (x , y , t )

(

where, f(z ) = tanh

( )

1 rz

z

r/ r2 / 4

1

)where, u¯ , v¯ and w¯ denotes the

mid-plane displacements in x, y and z directions, respectively. x & y represents the rotations of the mid-plane which accounts for the nonlinear variation of transverse shear stresses about x and y-axes, respectively. The shape function (Joshan et al., 2017) is f(z ) = (z ) + z , where, (z ) = tanh 1 (rz / ) and = (r / )/(1 r 2 /4) , r = 0.088. The relation between strain and displacement for the case of small strain and moderate rotations i.e. Von- Karman strain is expressed in Eq. (3):

2.1. Single layered S-FGM The single-layered S-FGM plate in which material properties are changing from metal phase at the bottom surface to ceramic phase at the top surface at different temperatures ranges in accordance with two power laws along the thickness is shown in Fig. 1(first column).

(0)

=

+z

= f (z )

(1) b

+ f (z )

(3.a)

(1) s

(0) s

(3.b)

where strain terms are expanded as given in Eq. (4):

2.2. S-FGM face sheets and homogenous hard core

(0)

The sandwich S-FGM plate with two face sheets of S-FGM, and a hard homogenous core at different temperatures ranges is shown in Fig. 1(second column). The material gradation in the face sheets is such as to ensure the continuously varying composition distribution and to avoid the interlaminar stresses which usually occurs in conventional sandwich plates. The effective material property for single layered and sandwich SFGM plate is computed in accordance with the simple rule of mixtures (ROM) for the ith layer as expressed in Eq. (1): (i) M(i) (z, T) = Mc V (i) t + Mm (1 V t )

Fig.

=

(0) (0) xx , yy ,

(0) xy

=

u¯ 1 + x 2

w¯ x

2

v¯ 1 + y 2

,

w¯ y

2

u¯ v¯ + + y x

,

w¯ x

w¯ y

(4.a)

(1)

ℳ m and ℳ c define the material properties of the constituent materials of FGM plate i.e. metal and ceramic, respectively. Vt is volume fraction of the constituent material and is given in Table 1. p denotes the inhomogeneity parameter ( 0 ) stating the shape of the profile for material property variation in the thickness direction. The geometry of the single layered and sandwich S-FGM plate is shown

(1) b

=

(1) (1) (1) b xx , b yy , b xy

=

(1) s

=

(1) (1) (1) s xx , s yy , s xy

=

(0) s

=

(0) xz ,

(0) yz

={

x,

2w ¯

x2 x

x y}

,

2w ¯

,

y2 y

y

,

,

2

x

y

+

2w ¯

x y

(4.b)

y

x

(4.c)

(4.d)

3.2. Stress-strain relation The linear constitutive relation for sandwich S-FGM plate is expressed as given in Eq. (5):

157

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Fig. 1. Effect of volume fraction exponent (p) and temperature differences (a) 0 K (b) 100 K (c) 300 K on Young's modulus of elasticity for single layered and (1-1-1) sandwich S-FGM plate arranged column-wise, respectively.

xx yy yz xz xy

Q11 Q12 0 0 0 Q12 Q22 0 0 0 0 Q66 0 0 = 0 0 0 0 Q66 0 0 0 0 0 Q66

expressed as,

T T

xx yy

E (z , T ) E (z , T ) E (z , T ) , Q12 = = = Q11 , Q66 2 2 2 1 1 1 E (z , T ) 1 = = Q11. 2(1 + ) 2

Q11 = Q22 =

yz xz

(5)

xy

where, { xx , yy, yz , xz , xy} and { xx , yy , yz , xz , xy } represents stresses and strains components, respectively and elastic stiffnesses are

(z ) is a thermal expansion coefficient and,

T (z) is the variation in

Table 1 Expression for the volume fraction of various layers of single layered and sandwich S-FGM plate. PLATE

Volume Fraction (Vt ) 0

S-FGM

1

0 1 2

Sandwich S-FGM

(

i z

(

0 0

i

0 1 2

1

i z i

0 0

) )

p

p

i

1

1 2

1 i

1

(

z i

1 1

)

p

1 1

)

p

1

1 2

(

z i

_____

2

2

_____

1

2

1

158

3

m

1 2

(

z i

2 2

)

p

m 1 2

(

3 z i

3 3

)

p

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Fig. 2. Geometry of (a) single layered and (b) sandwich (1-2-1) S-FGM plate.

Fig. 3. S-FGM Plate resting on two-parameter based elastic foundation with a description of generalized coordinates employed.

temperature across the thickness of the plate.

In terms of force (N), moments (M) and shear force resultants (Q), the above expression is simplified and given in Eq. (7):

3.3. Variational statements The Hamiltonian formulation is implemented to derive resulting governing equations which can be expressed in analytical form as given in Eq. (6):

Nxx

(0) xx

s Mxx

(1) s xx

U= A

(0) yy

+ Nyy s + Myy

+ (Q x

x

+ Nxy

(1) s yy

+ Qy

(0) xy

s + Mxy

b + Mxx (1) s xy

(1) b xx

q + (Mxx

(6)

b s Nxx Mxx Mxx

xx

+

yy

yy

+ 2(

xy

xy

+

yz

yz

+

h

xx

+

( )( ) ) dz w¯ x

w¯ x

2

xx z

2 2 xy

(

v¯ y

(

v¯ x

+

+

( )( ) ) dz w¯ y

u¯ y

w¯ y

+

w¯ w¯ x y

+

2

2 w¯ dz x2

2 w¯ dz y2

w¯ w¯ y x

2 yz 2

y dz

2

2 2

2 w¯ xy z x y dz

2 h

2 2

2

2

=

(1,

x

(z )

xz

2

dz +

x

y

dz +

xz

(z ))

yz

y

yy z

y

xx

(z )

yy

(z )

x

x

2

dz 2

+

h

2

(8.a)

(8.b)

dz+

y

y

dz+

2 x dz

dz

dz

2

2

2

xy z

yy xy

3.3.2. Work done The transverse load (qz ) acting on the top surface of the singlelayered and sandwich S-FGM plate results in storing potential energy

2

) dz + +

x

xx z

2

+

2

2

+

2

+

(1, z , (z ))

s Mxy

q Qx Mxx q Q y M yy

2 yy z

=

zx )] dv

zx

2

yy

A

u¯ x

2

2

U=

(

b Mxy

Nxy

After replacing the components of strain from Eq. (4) in the above equation, the expression for the virtual strain energy can be expressed as, 2

dxdy

xx

2

b s Nyy Myy Myy

3.3.1. Strain energy The virtual strain energy for FGM single layered and sandwich plate is given by, xx

y)

(1) b xy

where,

where U is the virtual strain energy, V is the virtual potential energy due to the elastic foundation, W is the virtual work done due to the transverse load, and K is the virtual kinetic energy.

[

q + M yy

b + Mxy

(7)

K ) dt = 0

t1

U=

x

(1) b yy

y)

t2

( U+ W+ V

b + Myy

xz

dxdy x dz

+

h

2

xy z

2

y

x

dz +

2 xy 2

159

2

(z )

y dz

yz

2

(z )

(

x

y

+

y

x

) dz

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

due to virtual work done is expressed as given in Eq. (9):

W=

qz wdxdy ¯

(9)

= J1 u¯

3.3.3. Potential energy The spring force or the reaction force ( ˆ e) in elastic foundation considered at the bottom surface of the single-layered and sandwich SFGM plate results in storing spring energy due to deformation of the foundation. This results in virtual potential energy stored in spring and expressed as given in Eq. (10):

ˆ e wdxdy ¯

V=

= J1 v¯

After replacing the components the displacement from Eq. (2), the above expression for virtual kinetic energy is expressed as given in Eq. (11):

K=

+ I2 A

(

w¯ x

+ J1 (u¯ J2

x

(

w¯ x

+

+

x

w¯ y

)

+

w¯ x

x

w¯ x

+

u¯ + v¯

w¯ y

w¯ y

+

v¯

y

+

+

w¯ y

2

)

+

y

w¯ y

y

)+J ( 3

x

x

+

y

z 2,

{1, z,

f(z ), z f(z ),

w¯ :

Nyy y

Nxy

+

2M b xx x2

= I0 v¯

x

2M b yy y2

(14.a)

v¯ =

I1 w¯ I0 y

J1 ¨ y I0

(14.b)

+

+2 +

=

I0 w¯ + I1 =

(

u¯ x

+ J2

v¯ y

+

(

¨x x

)

+

(

+

w¯ Nyy y

¨y y

(

)

2w¯

x2

x

(N

+ +

w¯ xx x

w¯ Nxy x 2w¯

y2

2M b xy

+2

(

(

I12

y

I2

I0 I1 J1 I0

+ qz +

x y

)(

(N

w¯ yy y

)( ¨x

2w ¯

x2

2w ¯

+ ¨y

+

x

+ Nxy

)

+ qz

w¯ y

(N + N ) )+q ˆ w¯ xx x

x

w¯ x

+ Nxy

y

y2

)

w¯ xy y e

z

) (15.a)

s b Mxy Mxx + + x y

b Mxy

J1 I1 I0

+ s Myy

y

+

b Myy

y

+

s Myx

x

+

q Mxx

y

b Myx

Qx = J3

J1 I1 I0

J12 ¨ x I0

w¯ x

J2

q M yy

x +

(12.b)

I2

2M b yy y2

+ J2 s Mxx + x

w¯ + J1 ¨ y y

x y

y

+

f(z ) 2} dz

(12.a)

2M b xy

(13)

x y

J1 ¨ x I0

I0 w¯ +

w¯ I1 + J1 ¨ x x I1

2

I1 w¯ I0 x

y)

The governing equations in the form of Euler- Lagrange equation in conjunction with the application of fundamental lemma of variational calculus is obtained after replacing Eq. (7), Eq. (9), Eq. (10) and Eq. (11) in Eq. (6) and simplify the equation term by term using integration by parts.

v¯:

Qy (12.e)

, Nxy =

x2

+

4. Field equations

u¯:

M q yy

x

u¯ =

2M b xx x2

2

Nxy Nxx + = I0 u¯ x y

2

, Ny =

v¯)

y

M b yx

+

x

Now, Substituting Eq. (14) in (12.c-12.e), the governing equations are reformulated as,

(11) where, {I0, I1, I2, J1, J2, J3} =

y2

dxdy

u¯ + v¯

x

w¯ x

w¯ y

w¯ x

M s yx

+

y

Replacing equation (13) into governing equation (12.a) and (12.b), we get,

2

(

(12.d)

w¯ + J3 ¨ y y

J2

2

Nx =

(u u + v v + w w ) dz dA

I1 u¯

Qx

The nonlinearity in the present formulation is due in-plane stretching as a result of moderate rotation which occurs due to large deformation. This type of nonlinearity is handled by introducing Airy's function (x , y , t ) and is given as (Dogan, 2013; Dinh Duc and Hong Cong, 2018; Duc et al., 2016),

2

I0 (u¯ u¯ + v¯ v¯ + w¯ w¯ )

q Mxx

y

4.1. Airy's function

(10)

3.3.4. Kinetic energy The virtual kinetic energy of the single-layered and sandwich S-FGM plate is given as,

A

M b yy

+

y

b Mxy

w¯ + J3 ¨ x x

J2

M s yy

y:

where, ˆ e = (Kw KP 2 ) w¯ , Kw and Kp are the spring stiffness and shear modulus of the elastic foundation, respectively.Kp provides more effective stiffness to the plate in comparison to Kw , since, former consider the shear effect which results in an increase in load-bearing capacity which results in an increase in stiffness.

K=

s b Mxy Mxx + + x y

s Mxx + x

x:

(15.b)

Q y = J3

J12 ¨ y I0

w¯ y

J2

(15.c)

4.2. Governing equations in terms of displacements Now, in terms of higher order strains, force and moment resultants are obtained after substituting Eq. (5) into Eq. (8) and given as:

)

b s Nxx Mxx Mxx

ˆe

b s Nyy Myy Myy = b s Nxy Mxy Mxy

)

1 2

1

1 (12.c) 160

12

13

21

22

23

( ) ( ) ( ) 1

2

1

11

AT

31

BT

1

2

32

1

2

33

A B F B D H F H J

FT

AT BT F T 0 0 0

(16.a)

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Qx Qy

=

q Mxx q M yy

(0)

( A + K) 2(1 + )

(0)

(16.b)

yz

4

(0) xz

( K + L) 2(1 + )

=

Substituting Eq. (4) in Eq. (19) and then into compatibility equation (18) results in,

xz

y4

(0) yz

=( =

(0) xx

+

(0) yy ), 12

(1) s xx

+

(1) s yy

(0) yy

+

(0) xx ),

+

(1) s xx

21

=(

23

=

(1) s yy

=

(0) xy ,

31

32

=

(1) b xy

(1) b xx

=

22

(1) s xx

+

(1) b yy

=

+ (

(1) s yy

+

(1) s xy , 33

+

+

12 ( x )

= I0 w¯ + +

¯) 21 (w

I12 I0

1

13 (

22 ( x )

y) 2w ¯

I2

x2

2BT

x2

1

+

+

+

(1) b yy

(1) b xx

+ (

(1) s yy ), 13

+

32 ( x )

+

33 (

2w ¯

+

y2 y)

2AT

BF A

¨x

I1 J1 I0

+ J2

y2

x2

x

¨y

+

y2

J2

w¯ x

y2

2w ¯

=A

2

x y

2w ¯

2w ¯

x2

y2

FT BT + x x

1 1

F+B A

5.1. Solutions assumed The solutions are assumed in such a way to satisfy the above-mentioned geometric boundary conditions. Thus, assumed solutions are expressed as trigonometric functions in terms of unknown coefficients (Wmn, xmn , ymn ) and are given as:

AT x

y)

J12

= J3

I0

¨ y + J1 I1 I0

J2

w¯ y

FT BT + y y

1 1

F+B A

w (x , y , t ) = AT y

2

2w ¯

2w ¯

x2

y2

y, t ) =

m=1

n=1

xmn

y (x ,

y, t ) =

m=1

n=1

ymn

(x , y , t ) =

1 (Nxx A

Nyy

B(

(1) b

(0) yy

=

1 (Nyy A

Nxx

B(

(1) b yy

(0) xy

=

1 2(1 + ) Nxy A

B

+

F

(1) s xx

(1) s xx )

(1) s yy )

+

(1) b xy

+

(1) s xy

F

(1) s yy

F

+ AT )

(1) s xy

n y b

m x a

n y b

m x a

n y b

(21)

1 1 Nx y 2 + Ny0 x 2 2 0 2 1mn (t )cos

2m x a

+

2 mn (t )cos

2n y b (22)

(18)

+ AT )

m x a

where, Nx 0 and Ny0 are equivalent axial load along x- and y-axis respectively to prevent the respective edges from moving. 1mn (t ) and 2 mn (t ) are the undetermined coefficients obtained after satisfying the compatibility equation (20) and making the simplified assumption that variation in temperature through the thickness of the S-FGM sandwich plate results in in-plane strains due to thermal stress resultant (N T ) that solely affects the stiffness of the plate and has no effect on the bending which occurs due to thermal moment resultants (MbT , MsT ) . The unknowns ( 1 (t ) and 2 (t ) : AT = 0) can be calculated by substituting Eq. (22) into Eq. (20):

Also, using Eq. (16a), we get,

=

( ) sin ( ) (t )cos ( ) sin ( ) (t )sin ( ) cos ( )

Wmn (t )sin

x (x ,

m = 1 n= 1

x y

(0) xx

n=1

+

2w ¯

=

m=1

The solution for airy stress function ( ) is also assumed in such a way so as to satisfy edge boundary condition and compatibility equation and is given as (Dinh Duc and Hong Cong, 2018; Duc et al., 2016; Dinh Duc et al., 2017),

The strain compatibility equation is now introduced

x y

2AT

Ny (x , 0) = Ny (x , b) = Ny0 ; Nx (0, y ) = Nx (a, y ) = Nx0

(17.a)

4.3. Strain compatibility equation

2 (0) xx

+

My (x , 0) = My (x , b) = Mx (0, y ) = Mx (a, y ) = 0

where, ij ’s are the symmetric linear operators that dictate the behavior of primary variables which includes transverse displacements and rotations in relation to the stiffness of the plate. ˜ is the nonlinear term representing the in-plane stretching of the plate. This results in cubic nonlinearity which causes hardening behavior of the S-FGM plate. ij ’s and ˜ are given in Appendix B.

2 (0) yy x2

x2

Neumann Conditions:

y

(17.c)

+

2AT

+

u¯ (x , 0) = u¯ (x , b) = 0; v¯ (0, y ) = v¯ (a , y ) = 0 w¯ (x , 0) = w¯ (x , b) = 0; w¯ (0, y ) = w¯ (a, y ) = 0 x (x , 0) = x (x , b) = 0; y (0, y ) = y (a , y ) = 0

2AT

+

J12 ¨ J1 I1 x + I0 I0

= J3

+

2 (0) xx y2

x4

The present formulation is solved for a plate for which all edges are simply supported with immovable edges. Thus, this combined condition of immovability and supports are fulfilled by considering Navier's solution in combination with some assumptions and will be discussed in following sections. The boundary conditions in accordance with the above-discussed condition are fulfilled by following well-designed boundary conditions on the boundaries of the plate: Dirichlet Conditions:

(17.b)

+

4

5. Method of exact solution

(1) s xx ),

+

ˆe + ˜

+ qz

+

¯) 31 (w

+

(20)

(1) s xy

=

2B T

+

23 (

y2

Exact solution of four governing differential equations obtained from (17) and (20) are now solved for single layered and sandwich SFGM plate using Navier's method with all edges simply supported.

The reduced stiffnesses and thermal stress resultants are defined in Appendix A. Eq. (4) is then substituted into Eq. (16) to obtain force and moment resultants in terms of displacements and then into the governing differential equation (15). The equation of motion in terms of transverse displacement, rotations and stress function (w¯ , x , y and ) is then given as,

¯) 11 (w

4

x2

(16.c)

where, 11

+2

(19.a) (19.b)

1mn (t )

(19.c) 161

=

AWmn (t ) 2 32µ2

2

(23.a)

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Fig. 4. Geometry defining temperature at the interfaces in the thickness direction and temperature profiles for different layers of plate.

2mn (t )

=

AWmn (t )2 µ2 32 2

where, µ =

m a

,

=

Tb1 (hm1) = Tb2 (hm1) = T2, Tb2 ( 1) = Tm ( 1) = T3, Tm ( 2) = Tt 1 ( 2)

(23.b)

= T4, Tt 1 (hm2) = Tb2 (hm2 ) = T5

n b

dTb1 dz

5.2. Expression for mechanical and thermal load

m=1 n=1

where,

{Qmn, T¯mn} =

4 ab

16Qa , mn 2

a b 0 0

m x n y {Qmn, T¯mn}sin sin a b

{qz (x , y ), T (x , y )}sin

(24)

n y b

Tb = T1 + (T2

T ( /2) = T3 and T (

=

(25)

m1

/2) = T1

/2) = T1

T2 =

z= 1

z= 2

z = hm2

(h 0

T3)

(26.a)

z < hm )

(hm

m1

(26.b)

h 0 + h1 2

z h0 hm h 0

(26.c)

+

5 i= 1

=

1 c2

(

1

)

z h 0 ip ( hm h 0

z h1 hm h1

(

1)i 5 i=1

1+

( )( )( 1 i 2

( )

k cm i km

k cm i km

1 1 + ip

)

( 1)i

) ( )( )

z h1 ip 1 i 2 hm h1

1 1 + ip

T1c2 (c3 + 1) + T3c1 (c4 + c5 + 1) c2 (c3 + 1) + c1 (c4 + c5 + 1)

• Sandwich S-FGM plate

Tb (hm) = Tt (hm) = T2

z = hm

dTt2 = dz

dTm dz

(26.d)

• S-FGM plate dT = t dz

z = hm2

0m

1 c1

+

and continuity conditions at the interface is given for.

dTb dz

z= 1

,

0m

• Sandwich S-FGM plate

T ( /2) = T6 and T (

T1)

where , hm =

subjected to boundary conditions for.

• S-FGM plate

dTm dz

=

z < h1)

The nonlinear temperature variation across the thickness of the FGM plate is based on a one-dimensional steady-state heat conduction equation. The temperature distribution across each layer is derived in accordance with the distribution of material properties. Since sigmoid function based sandwich plate has been proposed for the first time in literature, therefore the distribution of temperature across the thickness of sandwich S-FGM plate is derived first in accordance with the polynomial series method. In addition, temperature distribution for S-FGM plate based on steady state heat conduction equation is also not available in the literature. The 1D steady state heat conduction equation without heat source is given as,

=0

z=

dTt 1 , dz 2

Tt = T3 + (T2

5.3. Temperature variation in FGM plate

d dT k (z ) dz dz

z = hm1

dTb2 dz

• S-FGM plate

for the uniformly applied load on the top surface of the 16 T plate of magnitude Qa . Similarly T¯mn = mn 2i , for uniformly distributed temperature throughout the surface of the ith layer subjected to a thermal load of T .

Qmn =

,

The temperature at the boundaries and at the interfaces between two layers and other terms are presented in Fig. 4. On integrating twice Eq. (25), using boundary and continuity conditions and, implying a polynomial series method, the temperature variation across each layer of the sandwich S-FGM plate is obtained for.

( ) sin ( ) dx dy Where m x a

z = hm1

dTb2 dz

dTt1 = dz

The transverse mechanical and thermal load through the thickness is expanded in terms of double trigonometric series as,

{qz (x , y ), T (x , y )} =

=

Tb1 = T1 + (T2

T1)

Tb2 = T3 + (T2

z = hm

z < h1)

• Sandwich S-FGM plate 162

(h 0

0m

T3)

m1

(26.e)

(27.a)

z < hm1)

(hm1 (27.b)

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

T3h2 + z (T3 h1 h2

T4 h1

Tm =

T4)

z < h2)

T4)

2m

T6)

h3) h 0 + h1 h + h3 , hm 2 = 2 2 2

=

1 c2

2m

=

1 c2

m3

=

1 c1

m1

T2 = T3 = T4 = T5 =

+

z

1+ 1+

5 i=1

( ) ( )( )( ) ( ) ( 1) + ( ) ( ) ( )

z

1+

5 i=1

( )

z

1+

5 i=1

(

h0 hm1 h 0 z

h1 hm1 h1 h2 hm2 h2 h3 hm2 h3

h 0 ip ( hm1 h 0

1 i 1)i 2

z

k cm i km

k cm i km

h1 ip 1 i hm1 h1 2

) ( )( ) ) ( 1) ( ) ( ) ( ) (

( 1)i +

h2 ip 1 i hm2 h2 2

i i 1 2

k cm i km

1 1 + ip

(H f1 + H f2)(c1 (c4 +c5 +1) + c2 (c3 + 1)) + 2(c3 + 1)(c4 + 1)Hc

T1c1H f2 (c4 +c5 +1) + T6 (Hf1c1(c4 +c5 +1) + 2(c3 + 1)(c4 + 1)Hc + c2 (c3 + 1)(Hf1 + Hf 2)) (H f1 + H f2)(c1 (c4 +c5 +1) + c2 (c3 + 1)) + 2(c3 + 1)(c4 + 1)Hc

c2 = 1 +

( 1)i

h1

5 i=1

c5 = 1 +

5 i=1

1 i 2

k cm i km

1 1 + ip

k cm i km

i i 1 2

1 i 2

k cm i km

4

(28.b)

= f2mn 31 Wmn (t )

+

32 xmn (t )

33

(28.c)

s ) and mass matrix (Mij s ) are

ij

= 0; = 0;

(29)

u¯ x

=

1 (Nxx A

Nyy

B(

(1) b xx

+

(1) s xx )

F

(1) s xx )

1 2

( )

v¯ y

=

1 (Nyy A

Nxx

B(

(1) b yy

+

(1) s yy )

F

(1) s yy )

1 2

( )

w¯ 2 x

w¯ 2 y

(30)

T

A¯ mn 1

4

2 ) W (t )) mn

+

A 8(1

T

A¯ mn 1

2 ) W (t )) mn

2 ) ((B

2mn(1

2)

(µ2 +

4 2 mn(1

+

2)

A 8(1

2)

+ F )(µ

xmn (t )

ymn (t ))

B (µ2

ymn (t ))

B ( µ2

2 ) W (t ) 2 mn

((B + F )( µ

( µ2 +

+

xmn (t )

+

2 ) W (t ) 2 mn

(31) 5.6. Equation of motion for nonlinear dynamic analysis

1 1 + ip 5 i=1

=

+

),

i i 1 2

i

22 xmn (t )

ymn (t ) = f3mn

u dxdy x v dxdy y

Ny0 =

( ) ( 1) (1 + ( 1) ( ) ( ) ) ( 1) ( ) ( ) , c = 1 + ( 1) ( ) , ( )( )

5 i=1

c3 = 1 +

( )( )(

x

+

c1, c2, c3, c4, and c5 are the constants for both single layered and sandwich S-FGM plate given as, 5 i=1

=

Nx 0 =

(H f1 + H f2)(c1(c4 +c5 +1) + c2 (c3 + 1)) + 2(c3 + 1)(c4 + 1)Hc

c1 = 1 +

+

Substituting Eq. (30) into Eq. (29), the equivalent axial loads are obtained as,

T1H f2 (c2 (c3 + 1) + c1(c4 +c5 +1)) + T6 (2(c3 + 1)(c4 + 1)H c + (c1 (c4 +c5 +1) + c2 (c3 + 1))H f1)

h2 , Hc = h2

23 ymn (t )

21 Wmn (t )

Also, from equation (4) and (19), we get

T1(H f 2 (c2 (c3 + 1) + c1 (c4 +c5 +1)) + 2(c3 + 1)(c4 + 1)Hc) + T6Hf1 (c2 (c3 + 1) + c1(c4 +c5 +1))

h 0 , H f2 = h3

x

1 1 + ip

T1(c1H f2 (c4 +c5 +1) +c2 (c3 + 1)(H f1 + H f 2) + 2(c3 + 1)(c4 + 1)Hc) + T6c1H f1 (c4 +c5 +1) (H f1 + H f2)(c1(c4 +c5 +1) + c2 (c3 + 1)) + 2(c3 + 1)(c4 + 1)Hc

H f1 = h1

(28.a)

In order to implement immovability of the plate edges, in-plane deflections x and y are assumed zero in the average sense as,

1 1 + ip

z

nl Wmn

5.5. Equivalent axial loads to impose immovability

1 1 + ip

z

i

h3 ip hm2 h3 z

k cm i km

+

where elements of the stiffness matrix ( provided in Appendix C.

(27.f)

5 i=1

z

13 ymn (t )

¨mn (t ) + M32 ¨xmn (t ) + M33 ¨ y (t ) + M31 W mn

(27.e)

where, hm1 = =

+

(hm2

m3

+

¨mn (t ) + M22 ¨xmn (t ) + M23 ¨y (t ) + M21 W mn (27.d)

Tt2 = T6 + (T5

12 xmn (t )

11) Wmn (t )

11

(t ) 3

= Qmn + f1mn

(h2

z < h m2 )

0m

+

(27.c)

Tt1 = T4 + (T5

1 c1

¨mn (t ) + M12 ¨xmn (t ) + M13 ¨y (t ) + ( M11 W mn

(h1

Substituting Eq. (31) into the equation of motion (28) results in second-order nonlinear ordinary differential equation with quadratic and cubic nonlinearities as given in Eq. (32).

i i k cm km

k cm i km

¨mn (t ) + M12 ¨xmn (t ) + M13 ¨y (t ) + M11 W mn

The temperature distribution along the thickness for single layered and sandwich S-FGM is plotted using Eqs. (26) and (27), respectively. The temperature distribution is plotted for different temperature difference, as shown in Fig. 5, and for different volume fraction exponent, as shown in Fig. 6. The effect of the nonlinear distribution of temperature is more prominent at higher temperature difference. This can be confirmed from Fig. 1 where material properties are degraded abnormally at higher temperature difference.

+

13

ymn (t ) + ˜11

xmn (t ) W (t ) + ˜12

11 Wmn (t )

+

12 xmn (t )

ymn (t ) Wmn (t )

+ ˜13 Wmn (t )2 + ˜Wmn (t )3 = Qmn + f1mn (32.a)

¨mn (t ) + M22 ¨xmn (t ) + M23 ¨y (t ) + M21 W mn +

23 ymn (t )

+

The nonlinearity in the governing equation is solved using Airy's function in combination with the Galerkin method to obtain secondorder nonlinear ordinary differential equations. This is obtained by substituting Eqs. (21)–(24) in Eq. (17) and then applying the Galerkin method. This will results in duffing equation with hard spring characteristics which occur as a result of cubic nonlinearity present in the differential equation and given as:

33 ymn (t )

+

22 xmn (t )

(32.b)

= f2mn

¨mn (t ) + M32 ¨xmn (t ) + M33 ¨y (t ) + M31 W mn

5.4. Equation of motion

21 Wmn (t )

= f3mn

31 Wmn (t )

+

32 xmn (t )

(32.c)

where ˜11, ˜12, ˜13, ˜ are defined in Appendix C. Equation (32) is solved using the fourth order Runge-Kutta integration scheme. In differential equations, quadratic nonlinearities occur as a result of employing resistance to the edges from moving and cubic nonlinearities occur as a result of in-plane stretching and produce third-order harmonics. 163

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Fig. 5. Effect of the temperature difference on the variation of temperature through the thickness for single layered and sandwich S-FGM plate.

5.7. Equation of motion for free vibration analysis

rotary inertia is neglected. Thus, governing nonlinear differential equation (32) can be rewritten as,

The linear free vibration analysis is performed to compute the natural frequency of the system. This is done by eliminating the nonlinear part of Eq. (32) and setting:

(

2M ij

+

+ ˜12

Qmn = 0, fimn = =0 for i = 1, 2, 3. ij

11

T ) W (t ) mn

31 Wmn (t )

The static nonlinear transverse displacement is evaluated to perform convergence study for plate subjected to uniform thermo-mechanical loading. This is done by eliminating the mass matrix and governing nonlinear differential equation (32) are reformulated as,

+ ˜12

ymn (t ) Wmn (t )

+

13 ymn (t )

+ ˜13 Wmn

+ ˜11

xmn (t ) W

(34.a)

31 Wmn (t ) +

32

+

xmn (t ) +

= f2mn

(34.b)

ymn (t ) = f3mn

(34.c)

23 ymn (t ) 33

xmn (t ) Wmn (t )

+

22 xmn (t )

+

32 xmn (t )

+

23 ymn (t )

1Wmn (t )

33 ymn (t )

+

(35.b)

=0

x (t )

2 Wmn (t )

2

(35.c)

=0 and

y (t )

and substitute back into

+ ˜Wmn (t )3 =

Qmn

(36)

where, coefficients of the equations are defined in Appendix C. Assuming uniformly distributed harmonic excitation,

(t )

Qmn =

22 xmn (t )

+

¨mn (t ) + M11 W

(t ) 2

= Qmn + f1mn

+

+ ˜11

(35.a)

Solving 35(b) and 35(c) for 35(a), we get

+ ˜Wmn (t )3

21 Wmn (t )

13 ymn (t )

¨mn (t ) M11 W

+ Qmn =

(33)

=0

12 xmn (t )

+

+ ˜Wmn (t )3

5.8. Governing equation for nonlinear transverse displacement

+

12 xmn (t )

2 ymn (t ) Wmn (t ) + ˜13 Wmn (t )

21 Wmn (t )

11 Wmn (t )

+

16Qa sin( mn 2

nl t )

(37)

The displacement assumed is given as:

W (t) = wmax sin(

(38)

nl t )

Substituting Eqs. (37) and (38) in Eq. (36) and employing the Galerkin method to obtain an algebraic equation in terms of non-linear frequency ( nl ) .

The deflection and rotation are obtained by solving nonlinear coupled equation (34) using fsolve function of MATLAB (R2018b).

2 nl

5.9. Governing equation for nonlinear frequency-amplitude relation Harmonic balance method is used to derive a relationship between nonlinear frequency and maximum amplitude with an assumption that

as,

=

2 mn

1+

8wmax 3

2 1

+

2 3wmax ˜ 4 1

mn Qa M11 wmax

The relation between force amplitude and frequency ratio is written

Fig. 6. Effect of volume fraction exponent (p) on the variation of temperature through the thickness for single layered and sandwich S-FGM plate at ΔT = 300 K. 164

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

2

8wmax 3

1+

2

+

1

2 3wmax ˜ = 4 1

Qmax 2 mn wmax

a , mn = where, = nl , Qmax = mn M11 mn mn With no external force i.e. Qmax = 0;

Q

2 nl

=

where,

2 mn

1+

mn

8wmax 3

2 1

1

=

+

In this paper, only first four terms {(1, 1), (1, 3), (3, 1), (3, 3)} of the series are considered to characterize the transverse displacement. This is due to fact that the average error is less than 0.22% for all values of uniform loading which is acceptable for any computation. In addition, the inclusion of more number of terms is computationally more expensive at the cost of average error for dynamic analysis of plate.

(39)

16 2

2 3wmax ˜ 4 1

(40)

6.2. Validation studies

= Linear fundamental frequency of the system

M11

In this section, some examples are presented with the purpose of validating the present formulation for a single layered and sandwich FGM plate. The numerical results are exhibited in tabular form.

It is clearly seen from expression (40) that, the nonlinear frequency is displacement dependent. This is, of course, a well-known type of behavior for Duffing-type nonlinearities.

Example 1. This example is performed in order to validate the present formulation for various values of inhomogeneity parameter (p) at room temperature (300 K). The verification is performed for square Al/Al2O3 S-FGM plate, with all edges Simply Supported (SSSS) using Eq. (33). The results of Jung et al. (2016) for non-dimensional frequency ( ˜ = a2 m / Em / ) are compared with the results obtained using present formulation and tabulated in Table 5. The material property of constituent material is Em = 70 × 109 N /m2, m = 2702 Kg / m3 for aluminum (Al) and Ec = 380 × 109 N / m2 , c = 3800 Kg / m3 for alumina (Al2O3). The Poisson's ratio is constant and set as 0.3. It is observed from the comparison study that the present results are in good agreement for all inhomogeneity parameter (p), thickness ratio (a/ ) and elastic foundation parameters ( ¯ w , ¯ p) . Further, it is perceived that with an increase in inhomogeneity parameter, the difference in results also increases. This may occur as a result of shear strain function assumed in the present formulation as a trigonometric function whereas, Jung et al. (2016) assumed polynomial function. This naturally affects the accuracy of the results.

6. Computation and discussion of results The results for linear and nonlinear free vibration are computed for single layered and sandwich S-FGM plates that are resting on twoparameter elastic foundation. Various symmetric and unsymmetric sandwich S-FGM plates are analyzed and listed in Table 2. The dimensionless elastic foundation parameters are presented as:

¯w =

wa

D

4

, ¯p =

2 pa

D

,D=

Em 12(1

3

(41)

2)

Table 3 contains the temperature dependent material properties of constituent materials which are used by following functional relationship (Touloukian, 1967) to express effective material properties (ℳ) as a function of temperature: ℳ(c/m) = ℳ

(ℳ -1T

0

(i)

−1

+ 1+ ℳ 1 T(i)+ ℳ 2 T(i)2+ ℳ 3 T(i)3) (42)

where subscript (c/m) denotes constituent material of FGM plate is either ceramic (c) or metal (m), and subscript i denotes the layer of the single-layered or sandwich S-FGM plate. In addition, Temperature Independent material properties are evaluated from Eq. (41) at room temperature (300 K).

As mentioned earlier, no prior work has been done on the sandwich S-FGM plate based on the temperature distribution given by Eq. (26) to our best knowledge, which is our focus. Therefore, in the following examples, homogenous and/or P-FGM plate with volume fraction exponent, (p = 1) is considered.

6.1. Convergence study

Example 2. The accuracy of the present formulation in predicting higher order modes of vibration in a thermal environment (TD & TID) is evaluated in this example using Eq. (33). Results for the first two non2 )/ E dimensional fundamental frequencies ( ˜ = a2 0m (1 0m / ) for thick simply supported Si3N4/SUS304 S-FGM plate are compared with the results given by Huang et al. (Huang and Shen, 2004), where, 0m and E0m represents material properties of metal at room temperature. The material properties are provided in Table 3. Poisson's ratio is considered as 0.28. The results are in excellent agreement with an average error of 0.65% and 0.91% when compared with the solution based on TID and TD properties, respectively as exhibited in Table 6 for different temperature differences. The slight difference in results at higher temperature differences occurs due to different temperature profile employed in the present formulation.

In this section, convergence study is performed for the results of nonlinear transverse displacement for the various magnitude of uniform loading. The results for nonlinear transverse displacement is obtained for Al2O3 homogenous plate are compared with Azizian et al. (Azizian and Dawe, 1985). The material property of the constituent material is Ec = 380 × 109 N / m2 , = 0.3. The non-dimensional transverse displacement and the non-dimensional load are defined as 2 )/ E 4 , respectively. The w˜ = 200w¯ (a/2, b/2)/ a and Q¯a = 12Qa a4 (1 c convergence of solution is performed by including more number of terms (m, n) in calculating the transverse displacement. The terms signify the number of half sine waves in x and y-direction, respectively that should be included to give more accurate results. It is also noted that m and n are always odd integers since, the expression for uniform loading is identically zero for even integers. Thus, based on different combinations of m and n, non-dimensional transverse displacement is evaluated. Therefore, more the number of terms included in the solution, more will be the accuracy of the solution as depicted from Table 4.

Example 3. Next, the results of nonlinear frequency ratio ( nl / l ) obtained using present formulation (using Eq. (39)) in the thermal environment for various values of (wmax / ) being evaluated. The material properties are listed in Table 3. Poisson's ratio is considered

Table 2 Sandwich plate configurations in terms of fractions of different layers. Configuration

1

1-1-1 2-1-2 2-2-1 1-2-1 2-1-1

*.

1

=

6 10 10/3 4 4

/

1,

2

= /

Description

2

6 10 10 4

A plate having all layers of equal thickness Plate with core layer is half the thickness of facesheets Plate with the bottom layer is half the thickness of the top and core layer Plate with facesheets is half the thickness of the core layer Plate with the core layer and the bottom layer is half the thickness of the top layer

2

165

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Table 3 Temperature Dependent (TD) and Temperature Independent (TID) material properties of constituent materials for the FGM plate (Reddy and Chin, 1998). Materials

Properties

Temperature Dependent (TD) ℳ -1

ℳ

0

Temperature Independent (TID) ℳ

ℳ

1

2

ℳ

ℳ

3

(300K)

ZrO2

E(Pa) α (1/K) k(W/mK) ρ(Kg/m3)

0 0 0 0

244.27 × 109 12.766 × 10−6 0 0

−1.371 × 10−3 −1.491 × 10−3 0 0

1.214 × 10−6 1.006 × 10−5 0 0

−3.681 × 10-10 −6.778 × 10−11 0 0

168.063 × 109 18.591 × 10−6 1.77 3000

Si3N4

E(Pa) α (1/K) k(W/mK) ρ(Kg/m3)

0 0 0 0

348.43 × 109 5.8723 × 10−6 0 0

−3.070 × 10−4 9.095 × 10−4 0 0

2.160 × 10−7 0 0 0

−8.946 × 10−11 0 0 0

322.2715 × 109 7.4746 × 10−6 8.828 2370

SUS304

E(Pa) α (1/K) k(W/mK) ρ(Kg/m3)

0 0 0 0

201.04 × 109 12.330 × 10−6 0 0

3.079 × 10−4 8.086 × 10−4 0 0

−6.534 × 10−7 0 0 0

0 0 0 0

207.7877 × 109 15.321 × 10−6 12.14 8166

0.28. Results obtained for square Si3N4/SUS304 S-FGM plate, with all edges Simply Supported (SSSS) tabulated in Table 7 exhibits the comparability of the present formulation with the results of Huang et al. (Huang and Shen, 2004). The difference in results occurs mainly due to the fact that the present analysis is based on the harmonic balance method, while, Huang et al. (Huang and Shen, 2004) employed improved perturbation technique in order to determine nonlinear frequency ratio. In addition, the difference in temperature profile through the thickness also affects the results obtained.

the last section and it was observed that the present formulation provided results with good agreement with the previous studies. Now, in the following sections, parametric studies have been performed considering various geometric parameters, two-parameter based elastic foundation ( ¯ w , ¯ p) , thickness ratio (a/ ) , and inhomogeneity parameter (p). The material properties of ZrO2/SUS304 are considered for analysis in the subsequent sections and listed in Table 3. Poisson's ratio is constant through the thickness of the plate and considered as 0.3.

Example 4. The aim of this example is to verify and to predict the accuracy of the present formulation for different plate configurations at room temperature. The results are compared with Bennoun et al. (2016) for free vibration analysis ( = a2 0 / E0 / , 0 = 1Kg / m3, E0 = 1GPa) of Al/Al2O3 square sandwich S-FGM plate. The material property is defined as: Em = 70 × 109 N /m2 , m = 2707 Kg / m3 for aluminium (Al) and Ec = 380 × 109 N / m2 , c = 3800 Kg /m3 for alumina (Al2O3). Bennoun et al. (2016) obtain closed form solution using Navier's method for five variable refined plate theory. The results are presented in Table 8 which founds to be in consistent with Ref. (Bennoun et al., 2016) and maximum error is 0.49% for the various thickness ratio (a/ ) and plate configurations. The insignificant difference in results occurs due to thickness stretching effect which causes an increase in stiffness of the plate. Therefore, natural frequency for a plate having thickness stretching effect is always more in comparison to the plate without thickness stretching effect. This effect can simply be perceived from the results presented in Table 8.

6.3. Parameter studies

6.3.1. Benchmark results An extreme requirement of a material which could sustain the exposure to the harsh and severe thermal environment in the field of aerospace, automobile, and shipbuilding, has attracted a lot of researchers for the design and development of sandwich plate. Sandwich plates have an edge over its conventional predecessors due to their high strength and stiffness, low weight, and durability. Hence, the parametric study with different combinations are essentially required to check the dynamic properties of these plates. Moreover, most of the studies on sandwich FGM plates have considered power law (El Meiche et al., 2011; Thai et al., 2014; Zenkour, 2005; Nguyen et al., 2014; Natarajan and Manickam, 2012; Bessaim et al., 2013; Bennoun et al., 2016; Neves et al., 2013b) and exponential function (Sobhy, 2013; Meziane et al., 2014) but no such studies were conducted on the S- FGM sandwich plate in the thermal environment. This is because the effect of temperature cannot be neglected as far as the material with continuously graded mechanical properties is concerned. Therefore, the computed results are the benchmark for future studies on linear and nonlinear thermo-mechanical dynamic analysis of sandwich S-FGM plate.

The verification and prediction of the results computed for linear and nonlinear formulation for sandwich S-FGM plate are presented in

6.3.1.1. Effect of parameters on linear non-dimensional frequency parameter. Table 9 is tabulated to analyze the effect of thickness ratio

Table 4 Convergence and comparison of non-dimensional nonlinear transverse displacement for Al2O3 homogenous plate subjected to a uniform load. ( a/ = 20, ¯ w = ¯ p = 0)

#

Theory

Terms

Q¯a = 10 (#)

Q¯a = 50 (#)

Q¯a = 75 (#)

Q¯a = 100 (#)

Present

(1,1) (1,1) (3,1) (1,1) (3,1) (1,3) (3,3) (1,1) (3,1) (5,1) (1,3) (3,3) (5,3) (1,1) (3,1) (5,1) (1,3) (3,3) (5,3) (1,5) (3,5) (5,5) (1,1) (3,1) (5,1) (7,1) (1,3) (3,3) (5,3) (7,3) (1,5) (3,5) (5,5) (7,5)

0.4209(2.538) 0.4150(1.101) 0.4097(0.190) 0.4102(0.068) 0.4107(0.054) 0.4106(0.029)

2.0025(2.677) 1.9728(1.154) 1.9463(0.205) 1.9486(0.087) 1.9511(0.041) 1.9506(0.015)

2.8541(2.813) 2.8096(1.210) 2.7699(0.220) 2.7734(0.094) 2.7771(0.040) 2.7763(0.011)

3.5977(2.974) 3.5383(1.274) 3.4854(0.240) 3.49(0.109) 3.495(0.034) 3.4939(0.003)

Azizian et al. (Azizian and Dawe, 1985)

(3,4)∗

0.41048

1.9503

2.776

3.4938

Error (%), *(ns, n0), ns = number of strips, n0 = number of terms used in finite strip method. 166

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Table 5 Effect of thickness ratio (a/ ) and inhomogeneity parameter (p) on non-dimensional frequency ( ˜ ) for Al/Al2O3 S-FGM rectangular plate resting on Pasternak foundation. ¯w

0

¯p

a/

0

5 10 100

0

100

5 10 100

100

0

5 10 100

100

100

5 10 100

Volume fraction exponent (p) Theory

Homogenous

1

2

5

10

Present Jung et al. (Jung et al., 2016) Present Jung et al. (Jung et al., 2016) Present Jung et al. (Jung et al., 2016)

10.3761 10.3761 11.3351 11.3351 11.7315 11.7315

8.0292 8.0122 8.6883 8.6824 8.9550 8.9549

7.6588 7.6328 8.2376 8.2288 8.4687 8.4686

7.3220 7.2889 7.8312 7.8202 8.0321 8.0319

7.2181 7.183 7.7064 7.6948 7.8982 7.8981

Present Jung et al. (Jung et al., 2016) Present Jung et al. (Jung et al., 2016) Present Jung et al. (Jung et al., 2016)

15.1867 15.1868 15.9732 15.9732 16.3137 16.3138

14.4173 14.3818 14.9505 14.9401 15.1789 15.179

14.2076 14.152 14.6928 14.677 14.8973 14.897

14.0226 13.9491 14.4686 14.4478 14.6533 14.6535

13.9667 13.8879 14.4013 14.3792 14.5806 14.5804

Present Jung et al. (Jung et al., 2016) Present Jung et al. (Jung et al., 2016) Present Jung et al. (Jung et al., 2016)

10.6722 10.6723 11.6147 11.6147 12.0058 12.0058

8.4699 8.4517 9.1096 9.1035 9.3702 9.3702

8.1189 8.0912 8.6808 8.6716 8.9066 8.9066

7.8015 7.766 8.2961 8.2845 8.4926 8.4924

7.7039 7.6662 8.1784 8.1662 8.366 8.366

Present Jung et al. (Jung et al., 2016) Present Jung et al. (Jung et al., 2016) Present Jung et al. (Jung et al., 2016)

15.3904 15.3904 16.1728 16.1728 16.5121 16.5121

14.6668 14.6305 15.1993 15.1887 15.4277 15.4276

14.4604 14.403 14.946 14.93 15.15 15.15

14.2784 14.2029 14.7254 14.7043 14.9109 14.9107

14.2234 14.1425 14.6593 14.6368 14.839 14.839

Table 6 Comparison of first two non-dimensional linear frequency( ˜ ) for Si3N4/SUS304 S-FGM square plate. ( a/ = 8, a = 0.2, ΔT

Theory

¯ w = ¯ p = 0)

Mode (m, n) (1,1)

(1,2)

Si3N4

p=1

SUS304

Si3N4

p=1

SUS304

0

Huang et al. (Huang and Shen, 2004) Present

12.495 12.507

7.555 7.561

5.405 5.410

29.131 29.256

17.649 17.716

12.602 12.655

100a

Huang et al. (Huang and Shen, 2004) Present

12.382 12.449

7.514 7.505

5.335 5.352

29.243 29.197

17.694 17.659

12.587 12.596

300a

Huang et al. (Huang and Shen, 2004) Present Huang et al. (Huang and Shen, 2004) Present

12.213 12.332 12.397 12.406

7.305 7.392 7.474 7.479

5.104 5.232 5.311 5.334

28.976 29.078 29.083 29.061

17.486 17.545 17.607 17.584

12.342 12.475 12.539 12.553

Huang et al. (Huang and Shen, 2004) Present

11.984 12.235

7.171 7.319

4.971 5.127

28.504 28.726

17.213 17.324

12.089 12.238

100b 300b a b

TID. TD.

(a/ = 5,10) , volume fraction exponent (p = 1, 4), plate configurations (2-1-2, 2-1-1, 1-1-1, 2-2-1), and elastic foundation parameters ( ¯ w , ¯ p = 0&50) . The non-dimensional frequency parameter ( ˜ = a2 m / Em / ) is computed for ZrO2/SUS304 square S-FGM sandwich plate. It was found that as the core thickness becomes double (increases from 0.2 to 0.4 ), the non-dimensional frequency parameter increases for all values of foundation parameters and volume fraction exponent. This is due to the fact that, increase in core thickness for an S-FGM sandwich plate results in an increase in the stiffness of the plate. The dependency of material property on temperature significantly affects the natural frequency of the plate at higher temperature differences. It was observed that the average reduction in the non-dimensional natural frequency due to consideration of

dependency of material property is approximately 7.5% and 1.3% at a temperature difference of 300 K and 100 K, respectively in the absence of foundation. This reduction is due to a decrease in stiffness of a plate as result of (a) increase in temperature and (b) dependency of material property on temperature which results in deterioration of the material properties. In addition, with the consideration of foundation, the above percentage differences reduce to 1.46% and 0.263% at a temperature difference of 300 K and 100 K, respectively. It was observed from Table 9 that the increase in volume fraction exponent results in an increase in non-dimensional frequency for all configurations of sandwich S-FGM plate and temperature differences (TD & TID). Usually, an increase in volume fraction exponent results in an increase in the volume fraction of metal which is responsible for the 167

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

plate. This unexpected increase was due to a greater reduction in linear frequency due to the temperature difference, as discussed in the last section, in comparison to nonlinear frequency. Table 11 shows the effect of volume fraction exponent (p = 1, 10) on nonlinear frequency ratio ( nl / l ) for ZrO2/SUS304 square S-FGM sandwich plate of different configurations in a thermal environment. It was observed from the table that as volume fraction exponent increases, the nonlinear frequency ratio decreases for all configurations of sandwich S-FGM plate and temperature differences (TD & TID). This is due to the fact that an unexpected increase in linear frequency occurs with an increase in volume fraction exponent in comparison to a non-linear frequency which depends upon geometric stiffness. This makes overall frequency ratio term to decrease when volume fraction changes from 1 to 10. Table 12 shows the effect of aspect ratio (b/a = 0.5, 2) on nonlinear frequency ratio ( nl / l ) for ZrO2/SUS304 rectangle S-FGM sandwich plate of different configurations in a thermal environment. It was observed from the table that as aspect ratio increases, the nonlinear frequency ratio decreases for all configurations of sandwich S-FGM plate and temperature differences (TD & TID). As depicted earlier, this is due to the thickness reduction effect because, with an increase in aspect ratio from 0.5 to 2, relatively the plate thickness is reduced, or in other words, the slenderness of the plate increases. Hence, this results in a reduction of stiffness of S-FGM plate in the thickness direction. However, a considerable amount of reduction of nonlinear frequency ratio takes place for a plate at lower temperature difference in comparison to higher temperature difference as predicted from Table 12. Table 13 shows the effect of elastic foundation parameters ( ¯ w , ¯ p) on nonlinear frequency ratio ( nl / l ) for ZrO2/SUS304 square S-FGM sandwich plate of different configurations in a thermal environment. It was observed from the table that as parameters changes from Winkler foundation to Pasternak foundation, the nonlinear frequency ratio reduces drastically for all configurations of sandwich S-FGM plate for all configurations of sandwich S-FGM plate and temperature differences (TD & TID). This response is expected because the increase in stiffness of the S-FGM plate due to Pasternak foundation occurs as a result of transverse shear deformation of elastic springs consider in the foundation, whereas the Winkler foundation can be viewed as a series of independent closely spaced elastic springs and accounts for normal pressure only. Hence, this results in an excessive reduction in nonlinear frequency as Pasternak foundation lessens the effect of nonlinearity. This will make overall frequency ratio term to reduce drastically. However, there is a meagre effect of temperature on consideration of different foundation parameters for sandwich S-FGM plate. Table 10 - Table 13 shows that the nonlinear frequency ratio inO ( ). creases with an increase in maximum deflection ratio i.e. wmax This is due to the fact that linear frequency is independent of deflection and depends only on the mass and stiffness of the FGM plate. Therefore, nonlinear frequency increases substantially due to geometric stiffness of the plate without any considerable increment in linear frequency.

Table 7 The effect of amplitude ratio on nonlinear frequency ratio ( nl / l) for Si3N4/SUS304 single layered S-FGM square plate. ( a/ = 8, a = 0.2, ¯ w = ¯ p = 0, T = 100K ) wmax /

Theory

Si3N4

p=1

SUS304

0

Huang et al. (Huang and Shen, 2004) Present

1 0.999

1 0.999

1 0.999

0.2

Huang et al. (Huang and Shen, 2004) Present

1.022 1.019

1.022 1.028

1.022 1.019

0.4

Huang et al. (Huang and Shen, 2004) Present

1.084 1.081

1.084 1.098

1.082 1.081

0.6

Huang et al. (Huang and Shen, 2004) Present

1.181 1.175

1.18 1.201

1.172 1.176

0.8

Huang et al. (Huang and Shen, 2004) Present

1.303 1.297

1.301 1.329

1.296 1.299

1

Huang et al. (Huang and Shen, 2004) Present

1.446 1.437

1.442 1.475

1.438 1.441

Table 8 Comparison of the non-dimensional frequency parameter ( FGM square plate. (p = 1). Configuration

1-0-1 2-1-2 1-1-1 2-2-1 1-2-1 1-8-1

Theory

Present Bennoun 2016) Present Bennoun 2016) Present Bennoun 2016) Present Bennoun 2016) Present Bennoun 2016) Present Bennoun 2016)

) for Al/Al2O3 S-

(a/ ) 5

10

100

et al. (Bennoun et al.,

1.16981 1.17485

1.2432 1.2447

1.2716 1.27158

et al. (Bennoun et al.,

1.2235 1.22915

1.3001 1.30181

1.3297 1.32974

et al. (Bennoun et al.,

1.2714 1.2777

1.3533 1.35523

1.3851 1.38511

et al. (Bennoun et al.,

1.3079 1.31434

1.3957 1.39763

1.4299 1.42992

et al. (Bennoun et al.,

1.3468 1.35341

1.4393 1.44137

1.4755 1.47558

et al. (Bennoun et al.,

1.5246 1.53142

1.6489 1.65113

1.699 1.69906

reduction in stiffness and further reduces the fundamental frequency of the plate. But, this unpredicted increase occurs as a result of an increase in stiffness of the plate because SUS304 (metal) has a higher modulus of elasticity than ZrO2 (ceramic) at room temperature (300 K). 6.3.1.2. Effect of parameters on nonlinear frequency ratio for sandwich SFGM plate. Table 10 shows the effect of thickness ratio (a/ = 5,10) on nonlinear frequency ratio ( nl / l ) for ZrO2/SUS304 square S-FGM sandwich plate of different configurations in a thermal environment. It was observed from the table that as span-to-thickness ratio increases, the nonlinear frequency ratio decreases for all configurations of sandwich S-FGM plate and temperature differences (TD & TID). This was due to the thickness reduction effect because, with an increase in span-to-thickness ratio, the plate is transformed from thick to thin which results in a reduction of stiffness of S-FGM plate in the thickness direction. For thick plate (a/ = 5) with TD material property, nonlinear frequency ratio decreases with increase in temperature for all configurations of sandwich S-FGM plate as expected. However, for moderately thick (a/ = 10) and thick (a/ = 5) plate with TD & TID material property, the increase in nonlinear frequency ratio is observed with increase in temperature for all configurations of sandwich S-FGM

6.3.2. Effect of thermo-mechanical load on sandwich S-FGM plate Before proceeding to further sections for forced vibration analysis, it is essential to predict the effective load responsible for the commencement of the nonlinearity effect in the sandwich S-FGM plate. Thus, a transverse load of different amplitudes is applied to ZrO2/ SUS304 square S-FGM sandwich (1-1-1) plate. The non-dimensional transverse displacement and the non-dimensional load are defined as 2 )/ E 4 , respectively. The w˜ = 10w¯ (a/2, b/2)/ and Q¯a = 12Qa a4 (1 m results for transverse displacement is computed using linear (dashed line) and nonlinear (solid line) formulation as shown in Fig. 7. It was found that the effect of nonlinearity comes into view for non-dimensional load greater than 20, whereas, linear transverse displacement increases consistently with the increase in load and overpredicts the nonlinear transverse displacement. This is due to the effect of moderate rotation on transverse displacement considered in the nonlinear 168

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Table 9 Non-Dimensional fundamental frequency parameter for different configurations of ZrO2/SUS304 sandwich S-FGM square plates in the thermal environment. Configuration

(a/ )

p

(0,0)a

(50,50)a

TID

2-1-2

5 10

2-1-1

5 10

1-1-1

5 10

2-2-1

5 10

a

TD

TID

TD

ΔT = 100

ΔT = 300

ΔT = 100

ΔT = 300

ΔT = 100

ΔT = 300

ΔT = 100

ΔT = 300

1 4 1 4

6.3423 6.3678 6.8246 6.8731

6.2122 6.2386 6.3072 6.3618

6.2636 6.2979 6.7241 6.7838

5.8925 5.9389 5.6955 5.8034

13.5923 13.5873 14.0277 14.0456

13.5321 13.5276 13.7834 13.8026

13.5574 13.5608 13.9793 14.0028

13.3926 13.4016 13.5151 13.5554

1 4 1 4

6.3984 6.4206 6.8670 6.9085

6.2590 6.2813 6.3122 6.3545

6.3200 6.3720 6.7674 6.8238

5.9358 6.0000 5.6910 5.7968

13.7620 13.7581 14.1896 14.2048

13.6979 13.6939 13.9296 13.9438

13.7277 13.7646 14.1419 14.1669

13.5582 13.6014 13.6600 13.7025

1 4 1 4

6.5248 6.5459 7.0226 7.0620

6.3892 6.4114 6.4842 6.5289

6.4345 6.4590 6.9076 6.9561

6.0328 6.0680 5.7961 5.8853

14.0501 14.0449 14.5052 14.5193

13.9878 13.9831 14.2524 14.2676

14.0104 14.0048 14.4501 14.4683

13.8335 13.8324 13.9534 13.9855

1 4 1 4

6.6122 6.6304 7.0983 7.1311

6.4666 6.4849 6.5182 6.5518

6.5211 6.5570 6.9824 7.0271

6.1005 6.1468 5.8048 5.8904

14.3030 14.2989 14.7524 14.7639

14.2368 14.2323 14.4822 14.4930

14.2636 14.2817 14.6972 14.7157

14.0798 14.1027 14.1762 14.2087

( ¯ w , ¯ p)

Table 10 Effect of span-to-thickness ratio on nonlinear frequency ratio ( environment. ( p = 4, ¯ w = ¯ p = 0) Configuration

2-1-2

(a/ )

TID

5 10

TD

5 10

2-1-1

TID

5 10

TD

5 10

-1-1-1

TID

5 10

TD

5 10

2-2-1

TID

5 10

TD

5 10

ΔT

nl / l )

for different configuration of ZrO2/SUS304 sandwich S-FGM square plates in the thermal

(wmax / ) 0

0.5

1

1.2

1.5

2

100 300 100 300

1 1 1 1

1.1352 1.1403 1.1216 1.1402

1.4678 1.4838 1.4254 1.4834

1.6316 1.6524 1.5766 1.6519

1.8966 1.9245 1.8224 1.9238

2.3701 2.4096 2.2642 2.4088

100 300 100 300

1 1 1 1

1.1324 1.1344 1.1190 1.1435

1.4620 1.4762 1.4205 1.5070

1.6247 1.6449 1.5709 1.6853

1.8882 1.9175 1.8157 1.9725

2.3592 2.4042 2.2558 2.4833

100 300 100 300

1 1 1 1

1.1402 1.1460 1.1276 1.1488

1.4766 1.4941 1.4367 1.5012

1.6416 1.6642 1.5897 1.6731

1.9081 1.938 1.8380 1.9498

2.3836 2.4263 2.2834 2.4421

100 300 100 300

1 1 1 1

1.1380 1.1414 1.1257 1.1547

1.4716 1.48 1.4326 1.5285

1.6355 1.6580 1.5848 1.7102

1.9003 1.9322 1.8319 2.0022

2.3732 2.4210 2.2754 2.5201

100 300 100 300

1 1 1 1

1.1351 1.1404 1.1217 1.1407

1.4677 1.4839 1.4259 1.4849

1.6315 1.6525 1.5772 1.6538

1.8965 1.9247 1.8233 1.9265

2.3699 2.4099 2.2655 2.4125

100 300 100 300

1 1 1 1

1.1320 1.1341 1.1189 1.1453

1.4614 1.4765 1.4207 1.5142

1.6241 1.6455 1.5714 1.6949

1.8874 1.9187 1.8164 1.9858

2.3584 2.4063 2.2572 2.5027

100 300 100 300

1 1 1 1

1.1399 1.1457 1.1274 1.1489

1.4762 1.4938 1.4367 1.5022

1.6412 1.6639 1.5899 1.6744

1.9076 1.9381 1.8384 1.9518

2.3831 2.4263 2.2841 2.4452

100 300 100 300

1 1 1 1

1.1373 1.1407 1.1251 1.1565

1.4703 1.4877 1.4321 1.5364

1.6341 1.6580 1.5845 1.721

1.8988 1.9329 1.8318 2.0173

2.3715 2.4227 2.2758 2.5425

169

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Table 11 Effect of volume fraction exponent on nonlinear frequency ratio ( environment. ( a/ = 10, ¯ w = ¯ p = 0) Configuration

2-1-2

p

TID

1 10

TD

1 10

2-1-1

TID

1 10

TD

1 10

1-1-1

TID

1 10

TD

1 10

2-2-1

TID

1 10

TD

1 10

ΔT

nl / l )

for different configuration of ZrO2/SUS304 sandwich S-FGM square plates in the thermal

(wmax / ) 0

0.5

1

1.2

1.5

2

100 300 100 300

1 1 1 1

1.1233 1.1427 1.1212 1.1398

1.4309 1.4909 1.4243 1.4820

1.5838 1.6615 1.5752 1.6500

1.8322 1.9368 1.8206 1.9213

2.2782 2.4271 2.2616 2.4052

100 300 100 300

1 1 1 1

1.1209 1.1484 1.1187 1.1423

1.4269 1.5232 1.4192 1.5032

1.5793 1.7063 1.5692 1.6804

1.8272 2.0009 1.8134 1.9659

2.2725 2.5238 2.2525 2.4739

100 300 100 300

1 1 1 1

1.1283 1.1499 1.1274 1.1486

1.4403 1.5058 1.4360 1.5003

1.5946 1.6793 1.5888 1.6719

1.8450 1.9585 1.8367 1.9481

2.2938 2.4549 2.2814 2.4397

100 300 100 300

1 1 1 1

1.1265 1.1584 1.1255 1.1537

1.4371 1.5425 1.4318 1.5250

1.5910 1.7288 1.5836 1.7057

1.8407 2.0278 1.8302 1.9960

2.2887 2.5574 2.2728 2.5111

100 300 100 300

1 1 1 1

1.1231 1.1427 1.1215 1.1403

1.4303 1.4909 1.4250 1.4837

1.5830 1.6615 1.5761 1.6523

1.8311 1.9368 1.8218 1.9244

2.2766 2.4271 2.2634 2.4096

100 300 100 300

1 1 1 1

1.1205 1.1494 1.1186 1.1443

1.4259 1.5276 1.4197 1.5110

1.5781 1.7123 1.5700 1.6908

1.8257 2.0092 1.8146 1.9802

2.2705 2.5360 2.2545 2.4947

100 300 100 300

1 1 1 1

1.1281 1.1498 1.1273 1.1487

1.4396 1.5059 1.4362 1.5015

1.5938 1.6794 1.5891 1.6735

1.8439 1.9587 1.8373 1.9505

2.2923 2.4554 2.2825 2.4433

100 300 100 300

1 1 1 1

1.1259 1.1598 1.1249 1.1556

1.4358 1.5482 1.4314 1.5335

1.5895 1.7366 1.5834 1.7171

1.8390 2.0385 1.8304 2.0120

2.2865 2.5732 2.2736 2.5349

formulation for sandwich S-FGM plate.

1-2 configuration with ΔT = 100 K is as shown in Fig. 8. The dynamic response shows the quasi-periodicity of the system at a/ = 5 as the system shows bounded trajectories with multiple loops (a kind of net structure); also, discrete points/loops are arranged in the Poincaré map as shown in Fig. 8(a). With an increase in the span-to-thickness ratio as a/ = 10 , the system gains its periodicity with two strange attractors and multiple loops are formed with large amplitudes as shown in Fig. 8(b). The time displacement confirms the weak quasi-periodicity with some irregular peaks in the time response. Hence, with this plate configuration, the variation of the span-to-thickness ratio shows the route to the quasi-periodic nature of the system as with a/ = 5 (quasiperiodic), to a/ = 10 (weak quasi-periodic). With the increase in temperature at ΔT = 300 K having same plate configurations, the dynamic response shows the weak chaotic behavior of the system at a/ = 5 as the system shows dense bounded trajectories with cluster form of multiple loops; also, dense discrete points/loops are arranged in the Poincaré map as shown in Fig. 9(a). With an increase in the span-to-thickness ratio as a/ = 10 , the system shows quasi-periodicity with two strange attractors as shown in Fig. 9(b). The time displacement confirms the quasi-periodicity with some irregular peaks in the time response. Hence, with this plate configuration, the variation of the span-to-thickness ratio shows the route to the quasiperiodic nature of the system as with a/ = 5 (quasi-periodic), to a/ = 10 (weak quasi-periodic).

6.3.3. Time domain analysis Nonlinear time domain behavior is critically important for gaining a comprehensive understanding of the complex nonlinear phenomena of FGM plate dynamics and is best carried out employing time displacement responses, phase-plane plots and Poincaré maps, which are arranged column-wise in the plots, respectively. The plate under the thermal environment is subjected to harmonic excitation on the top surface with the uniformly distributed non-dimensional load (Q¯a = 50) and bi-sinusoidal thermal load (ΔT = 0 K, 100 K, 300 K). The TD material properties are considered for analysis and tabulated in Table 3. The plate is subjected to modal excitation which is calculated using Eq. (33) and dynamic responses are obtained using Eq. (32). 6.3.3.1. Effect of span-to-thickness ratio. The S-FGM sandwich (2-1-2) plate is examined for dynamic behavior by considering the very thick and thick plate as shown in Fig. 8 and Fig. 9 for thermal load of 100 K and 300 K, respectively. The modal frequency for thick (a/ = 5) and very thick (a/ = 10) plate is 6397 rad/s (1018.1 Hz) and 3516 rad/s (559.58 Hz) respectively for ΔT = 100 K, and 6170 rad/s (981.98 Hz) and 3318 rad/s (528.1 Hz) respectively for ΔT = 300 K. When moving from very thick to the thick plate, the stiffness of the plate reduces to a great extent. This occurs consequently due to a reduction in thickness of the plate. The dynamic responses of the S-FGM sandwich plate with a 2170

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Table 12 Effect of aspect ratio on nonlinear frequency ratio ( ( p = 4, a/ = 10, ¯ w = ¯ p = 0) Configuration

2-1-2

b/a

TID

0.5 2

TD

0.5 2

2-1-1

TID

0.5 2

TD

0.5 2

1-1-1

TID

0.5 2

TD

0.5 2

2-2-1

TID

0.5 2

TD

0.5 2

nl / l )

ΔT

for different configuration of ZrO2/SUS304 sandwich S-FGM square plates in the thermal environment. (wmax / ) 0

0.5

1

1.2

1.5

2

100 300 100 300

1 1 1 1

1.1557 1.1647 1.1519 1.1933

1.5305 1.5576 1.5190 1.6420

1.7126 1.7475 1.6979 1.8555

2.0051 2.0516 1.9854 2.1947

2.5238 2.5894 2.4960 2.7902

100 300 100 300

1 1 1 1

1.1523 1.1590 1.1495 1.2180

1.5237 1.5536 1.5154 1.7317

1.7045 1.7450 1.6940 1.9732

1.9952 2.0520 1.9812 2.3548

2.5112 2.5951 2.4916 3.0204

100 300 100 300

1 1 1 1

1.1620 1.1720 1.1597 1.2076

1.5414 1.5711 1.5339 1.6725

1.7250 1.7630 1.7152 1.8919

2.0193 2.0699 2.0060 2.2398

2.5407 2.6118 2.5215 2.8490

100 300 100 300

1 1 1 1

1.1593 1.1682 1.1582 1.2388

1.5354 1.5689 1.5313 1.7730

1.7177 1.7622 1.7123 2.0218

2.0101 2.0714 2.0027 2.4137

2.5285 2.6177 2.5177 3.0956

100 300 100 300

1 1 1 1

1.1557 1.1649 1.1522 1.1945

1.5306 1.5582 1.5200 1.6455

1.7128 1.7482 1.6991 1.8598

2.0053 2.0525 1.9870 2.2005

2.5241 2.5907 2.4982 2.7983

100 300 100 300

1 1 1 1

1.1520 1.1591 1.1496 1.2251

1.5233 1.5552 1.5163 1.7540

1.7042 1.7475 1.6953 2.0019

1.9949 2.0556 1.9832 2.3931

2.5110 2.6007 2.4947 3.0745

100 300 100 300

1 1 1 1

1.1617 1.1718 1.1596 1.2083

1.5411 1.5712 1.5343 1.6752

1.7247 1.7632 1.7157 1.8956

2.0191 2.0703 2.0069 2.2448

2.5406 2.6126 2.5231 2.8564

100 300 100 300

1 1 1 1

1.1585 1.1678 1.1577 1.2474

1.5343 1.5701 1.5314 1.8005

1.7165 1.7642 1.7127 2.0573

2.0089 2.0747 2.0037 2.4611

2.5273 2.6232 2.5198 3.1627

plate. This is due to the inhomogeneity that plays a vital role in affecting the material properties of sandwich S-FGM plate. The dynamic responses of the S-FGM sandwich plate with 2-1-2 configuration with ΔT = 100 K for volume fraction exponent as ceramic is as shown in Fig. 11. The response shows the chaotic of the system with some dense bounded trajectories. Also, dense discrete points/loops are so arranged in the Poincaré map that these indicate the chaotic nature as shown in Fig. 11(a). With an increase in the volume fraction exponent (p = 2), the system losses its chaotic nature with two attractors as shown in Fig. 11(b).

6.3.3.2. Effect of aspect ratio. The nonlinear dynamic responses for the (2-1-2) sandwich S-FGM plate is analyzed as depicted from Fig. 10. The modal frequency obtained for (b/a = 0.5) and (b/a = 2) plate is 8347 rad/s (1328.46 Hz) and 2217 rad/s (352.85 Hz), respectively for ΔT = 100 K. The reduction in stiffness of the plate occurs with an increase in aspect ratios. This is due to the fact that overall dimensions of the plate increase for same thickness ratio. The dynamic responses of the S-FGM sandwich plate with a 2-1-2 configuration with ΔT = 100 K for aspect ratios b/a = 0.5 is as shown in Fig. 10. The response shows the quasi-periodicity of the system with some dense bounded trajectories. Also, discrete points/loops are so arranged that these indicate the onset of chaos in the Poincaré map as shown in Fig. 10(a). With an increase in the aspect ratios b/a = 2, the system shows a chaotic nature with a strong attractor as shown in Fig. 10(b). The time displacement confirms the irregular peaks indicate chaotic nature. Hence, with this plate configuration, the variation of the aspect ratios shows the route to chaotic nature of the system as with b/a = 0.5 (quasi-periodic) to b/a = 2 (chaotic).

6.3.3.4. Effect of elastic foundation parameter. The effect of elastic foundation parameters on the dynamic behavior of the S-FGM sandwich (2-1-2) plate is analyzed as shown in Fig. 12. The natural frequency obtained for sandwich plate resting only on Winkler foundation (K¯ w = 100, K¯ p = 0) and on Pasternak foundation (K¯ w = 0, K¯ p = 100) is 2003 rad/s (318.8 Hz) and 4638 rad/s (738.2 Hz), respectively for ΔT = 100 K. The effect of Pasternak foundation predominates the Winkler foundation for the sandwich SFGM plate as significant change is observed while considering Pasternak foundation. This is due to the fact that Pasternak foundation incorporates transverse shear deformation of elastic springs which results in an increase in stiffness of the S-FGM plate, unlike the Winkler foundation which is simulated as a series of independent closely spaced elastic springs and accounts for normal

6.3.3.3. Effect of volume fraction exponent. The effect of different aspect ratios on the dynamic behavior of the S-FGM sandwich (2-1-2) plate is analyzed as shown in Fig. 11. The natural frequency obtained for ceramic and (p = 2) sandwich plate is 2049 rad/s (326.1 Hz) and 1749 rad/s (278.4 Hz), respectively for ΔT = 100 K,. The transition of fully ceramic plate to FGM plate results in a reduction of stiffness of the 171

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Table 13 Effect of elastic foundation parameters on nonlinear frequency ratio ( environment. ( p = 4, a/ = 10) Configuration

2-1-2

( ¯ w , ¯ p)

TID

(100,0) (0,100)

TD

(100,0) (0,100)

2-1-1

TID

(100,0) (0,100)

TD

(100,0) (0,100)

1-1-1

TID

(100,0) (0,100)

TD

(100,0) (0,100)

2-2-1

TID

(100,0) (0,100)

TD

(100,0) (0,100)

ΔT

nl / l )

for different configuration of ZrO2/SUS304 sandwich S-FGM square plates in the thermal

(wmax / ) 0

0.5

1

1.2

1.5

2

100 300 100 300

1 1 1 1

1.0943 1.1052 1.0182 1.0185

1.3380 1.3732 1.0709 1.0723

1.4622 1.5085 1.1006 1.1026

1.6668 1.7300 1.1534 1.1563

2.0399 2.1314 1.2598 1.2645

100 300 100 300

1 1 1 1

1.0918 1.1026 1.0174 1.0162

1.3322 1.3753 1.0685 1.0654

1.4551 1.5135 1.0974 1.0934

1.6578 1.7399 1.1487 1.1434

2.0280 2.1502 1.2525 1.2447

100 300 100 300

1 1 1 1

1.0987 1.1109 1.0189 1.0193

1.3461 1.3849 1.0721 1.0736

1.4716 1.5223 1.1020 1.1041

1.6778 1.7469 1.1549 1.1580

2.0532 2.1529 1.2615 1.2665

100 300 100 300

1 1 1 1

1.0967 1.1098 1.0182 1.0171

1.3410 1.3889 1.0698 1.0670

1.4652 1.5291 1.0989 1.0952

1.6695 1.7583 1.1504 1.1453

2.0418 2.1726 1.2545 1.2467

100 300 100 300

1 1 1 1

1.0941 1.1051 1.0180 1.0183

1.3373 1.3729 1.0700 1.0714

1.4613 1.5080 1.0994 1.1014

1.6655 1.7294 1.1516 1.1545

2.0381 2.1305 1.2570 1.2617

100 300 100 300

1 1 1 1

1.0913 1.1027 1.0171 1.0158

1.3311 1.3767 1.0674 1.0641

1.4537 1.5156 1.0959 1.0917

1.6560 1.7431 1.1465 1.1408

2.0255 2.1552 1.2490 1.2406

100 300 100 300

1 1 1 1

1.0982 1.1104 1.0186 1.0189

1.3449 1.3839 1.0710 1.0725

1.4700 1.5211 1.1005 1.1026

1.6758 1.7453 1.1527 1.1558

2.0505 2.1508 1.2580 1.2630

100 300 100 300

1 1 1 1

1.0957 1.1095 1.0177 1.0166

1.3390 1.3899 1.0684 1.0653

1.4627 1.5309 1.0970 1.0930

1.6665 1.7613 1.1477 1.1421

2.0379 2.1777 1.2502 1.2417

response shows the intermittent chaotic nature of the system with some intermediate bursts in trajectories and in time displacement response. Also, dense discrete points/loops are so arranged in the Poincaré map that show intermittency in Fig. 12(a). With different elastic foundation parameter K¯ w = 0, K¯ p = 100 , the system shows quasi-periodic nature with attractor as shown in Fig. 12(b). The time displacement confirms the quasi-periodicity with some irregular peaks in the time response. 6.3.3.5. Effect of plate configurations. The effect of plate configuration on the dynamic behavior of the S-FGM sandwich plate at ΔT = 600 K is analyzed as shown in Fig. 13 and Fig. 14 for the symmetric and nonsymmetric plate, respectively. The natural frequency for symmetric sandwich plate (2-1-2) and (1-1-1) is 3104 rad/s (494 Hz) and 3179 rad/s (505.95 Hz), respectively. Also, the natural frequency for non-symmetric sandwich plate (2-1-1) and (2-2-1) is 3213 rad/s (511.4 Hz) and 3221 rad/s (512.6 Hz), respectively. The increase in natural frequency from (2-1-2) to (1-1-1) or from (2-1-1) and (2-2-1) occurs as a result of an increase in core thickness of the plate which is a homogenous ceramic material. The dynamic responses of the S-FGM sandwich plate with the symmetric configuration at ΔT = 600 K is as shown in Fig. 13. The response shows the chaotic nature of the system with some intermediate bursts in trajectories and in time displacement response. Also, dense discrete points/loops are so arranged in the Poincaré map that shows chaotic nature in for 2-1-2 plate configuration as Fig. 13(a). With 1-1-1 plate configuration at ΔT = 600 K, the system

Fig. 7. Variation of non-dimensional transverse displacement with non-dimensional load for ZrO2/SUS304 (1-1-1) sandwich S-FGM square plate ( p = 4, a/ = 10) (____nonlinearity, … …. linearity).

pressure only. The dynamic responses of the S-FGM sandwich plate with 2-1-2 configuration with ΔT = 100 K for different elastic foundation parameter K¯ w = 100, K¯ p = 0 is as shown in Fig. 12. The 172

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Fig. 8. Effect of the span-to-thickness ratio (a) a/ = 5 (b) a/ = 10 on nonlinear dynamic responses for (2-1-2) ZrO2/SUS304 S-FGM sandwich square plate at ΔT = 100 K. (p = 4, K¯ w = K¯ p = 0)

shows a chaotic nature with large amplitude as shown in Fig. 13(b). The time displacement confirms the chaotic nature with irregular peaks. Hence, with this symmetric plate configuration, the chaotic nature dominates ΔT = 600 K. While, the nature of the dynamic response is changed to quasi-periodic for non-symmetric plate configuration as shown in Fig. 14.

to chaos with varying thermal load. At ΔT = 0 K, a multi-orbit periodic nature is observed with five discrete point loops in Poincaré map (Fig. 15(a)), while the change in temperature up to 100 K and 300 K, the system exhibits quasi-periodic nature with dense outer trajectories (Fig. 15(b and c)). Hence, with an increase in temperature, the system loses its periodicity to leading towards chaotic nature. Which is clearly observed ΔT = 600 K (Fig. 15(d)), as dense discrete points in the Poincaré map that show chaotic nature.

6.3.3.6. Effect of thermal load. The effect of thermal load on the dynamic behavior of the S-FGM sandwich plate is analyzed as shown in Fig. 15 for (2-1-2) plate. The natural frequency corresponding to thermal load 0 K, 100 K, 300 K and 600 K is 3620 rad/s (576.14 Hz), 3572 rad/s (568.5 Hz), 3441 rad/s (547.7 Hz), 3105 rad/s (494.2 Hz), respectively. As discussed earlier, the decrease in natural frequency occurs as a result of an increase in temperature from ΔT = 0 K to ΔT = 600 K. This is due to degradation of material property with the rise in temperature. The dynamic responses of the sandwich square plate (2-1-2) at (a) ΔT = 0 K (a) ΔT = 100 K. (a) ΔT = 300 K (a) ΔT = 600 K as shown in Fig. 15. The response clearly shows the route

7. Conclusions In this study, a new sigmoid law based sandwich functionally graded plate (S-FGM) resting of Pasternak elastic foundation was analyzed for evaluating the nonlinear vibration characteristics in the thermal environment. The displacement field is defined using nonpolynomial based inverse hyperbolic shear deformation theory. The nonlinear rise in temperature is considered across the thickness due to the temperature difference between the top and the bottom surface of

Fig. 9. Effect of the span-to-thickness ratio (a) a/ = 5 (b) a/ = 10 on nonlinear dynamic responses for (2-1-2) ZrO2/SUS304 S-FGM sandwich square plate at ΔT = 300 K. (p = 4, K¯ w = K¯ p = 0) 173

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Fig. 10. Effect of aspect ratios (a) b/a = 0.5 (b) b/a = 2 on nonlinear dynamic responses for (2-1-2) ZrO2/SUS304 S-FGM sandwich plate at ΔT = 100 K. (p = 4, a/ = 10, K¯ w = K¯ p = 0)

Fig. 11. Effect of volume fraction exponent (a) ceramic (b) p = 2 on nonlinear dynamic responses for (2-1-2) ZrO2/SUS304 S-FGM sandwich square plate at ΔT = 100 K. (a/ = 20, K¯ w = K¯ p = 0)

the simply supported plate with immovable edges. The Hamiltonian formulation was done to derive governing equations and nonlinearity (due to Von- Karman strains) is worked out using Airy's function in conjunction with Galerkin's method. The time and frequency domain analysis is then performed using a numerical integration scheme and harmonic balance method, respectively. Wide-Ranging parametric studies for, linear and nonlinear, frequency and time domain analysis have been performed by taking into consideration the effect of thickness ratio, inhomogeneity parameter, thermal load, and foundation parameter for various configurations of the sandwich plates. Poincaré maps, phase-plane plots and time responses are demonstrated to study the periodicity of sandwich S-FGM plate under harmonic excitation. The results obtained on the basis of this formulation are compared with existing theory solutions based on other shear deformation theories.

We note the following important and interesting findings from our study:

• The dependency of material property on temperature significantly • •

174

affect the natural frequency of the plate at higher temperature differences. The sandwich S-FGM plate of different configurations is found to have an increase in non-dimensional frequency parameter for all values of foundation parameters and volume fraction exponent, as the core thickness increases from 0.2 to 0.4 . The unexpected increase in linear frequency occurs with an increase in volume fraction exponent in comparison to a non-linear frequency which depends upon geometric stiffness. This makes overall frequency ratio term to decrease with increase in volume fraction

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Fig. 12. Dynamic responses for different elastic foundation parameter (a) K¯ w = 100, K¯ p = 0 (b) K¯ w = 0, K¯ p = 100 of ZrO2/SUS304 S-FGM sandwich square plate (2-12) at ΔT = 100 K. (p = 4, a/ = 20)

exponent.

• The Pasternak foundation is more effective in comparison to the •

Winkler foundation. An increase in the Pasternak foundation parameter results in creating a near resonance condition which in turn results in multiple steady state solutions. It is found that the effect of nonlinearity comes into view for nondimensional load greater than 20, whereas, linear transverse displacement increases consistently with the increase in load and

• •

overpredicts the nonlinear transverse displacement. This is due to the effect of moderate rotation on transverse displacement considered in the nonlinear formulation for sandwich S-FGM plate. The variation of aspect ratios shows the route to chaos. With the Winkler foundation, the response is chaotic but becomes weakly chaotic with the introduction of the Pasternak type foundation. The dynamic response clearly shows the route to chaos with the varying thermal load from ΔT = 0–600 K.

Fig. 13. Dynamic responses for different symmetric (a) 2-1-2 (b) 1-1-1 ZrO2/SUS304 S-FGM sandwich square plate at ΔT = 600 K. (p = 4, a/ = 20, K¯ w = K¯ p = 50)

175

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Fig. 14. Dynamic responses for (p = 4, a/ = 20, K¯ w = K¯ p = 50)

different

non-symmetric

(a)

2-1-1

(b)

2-2-1

ZrO2/SUS304

S-FGM

sandwich

square

plate

at

ΔT = 600 K.

Fig. 15. Effect of thermal load (a) ΔT = 0 K (a) ΔT = 100 K. (a) ΔT = 300 K (a) ΔT = 600 K for nonlinear dynamic analysis of ZrO2/SUS304 S-FGM sandwich square plate. (p = 4, a/ = 20, K¯ w = K¯ p = 50)

176

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.euromechsol.2019.04.005. Appendix A. Reduced stiffnesses and thermal stress resultants i

E ( z , T ) {1, z, z 2,

{A, B , D, F , H , J , K , L} =

(z ) 2 ,

(z ), z (z ),

(z ),

i 1

i

{AT , BT , F T } =

( z, T ) E ( z, T ) T {1, z,

(z )} dz

for i = 1, 2, 3, 4, 5.

i 1 i

1

{N T , MbT , MsT } =

( z, T ) E ( z, T ) T {1, z,

1

(z )} dz

i 1

Appendix B. Linear and nonlinear operators

11 12

13

21

22

2

1

1

=

2

1 1

=

˜ =

(

1

=

2

1

( (

BF A BF A

2

2w ¯ y2 x 2

2

(

1

=

2)

(1

)(

+

B2 A

+

B2 A

23

=

31

=

1 2 (1 1

2)

1 2 (1

32

=

33

= J

J

)

(1

F2

(

B2 A

F2 A

J

) F2 A

(H

+

1 (A 2 2 ( + 1)

y2 x2

x

x3 3

y y3

)(

(

BF A

+ 2 K + L)

x

BF A

+

)

3

3

+

x

x y2

)

3

y y x2

+

2w¯ 2

+

)

(H

BF A

+

4w¯ x 2 y2

)( + H))

(D + H )

+

A

+2

y4

(D

(H

1 (A 2 2 (1 + )

4w¯

+

x4

B2 A

+

+

4w¯

(D + H )

2 2w¯ x y x y

BF A

F2 A

= J

D

B2 A

BF A

D

(H )

BF A

+ 2K + L )

3w¯

(

BF A

BF A

BF A

+

B2 A

3w¯ x2 y

+

B2 A

+

)

(D + H )

+

y3

)

(

BF A

B2 A

(

)(

3w¯ x y2

+

x3

+

)

H

3w ¯

)(

2

1 2)

(1

(D + H )

)

(D + H )

)

x

+

2 1 x 2 (1 + ) y 2

y

+

2 1 y 2 ( + 1) x 2

x2

2

y

x y

)

B2 A

(D + H )

)

2

2

1 (1

x

x y

2)

y2

y

Appendix C. Elements of mass and stiffness matrix

M11 =

(

M12 = M13 =

11

=

12

=

13

=

nl

=

)

I12

I2 (

I0

(J (J

2

I1 J1 I0

2

I1 J1 I0

1 2

1

(

µ 1

2

1

2

A (µ 4 16

+

2)

I0

M21 =

) )

M22 =

D (µ2 +

2 )2

2

(

J12 I0

Kp (µ2 +

Kw

) + H ) ) (µ

BF A

+

B2 A

(D + H ) (µ2 +

BF A

+

B2 A

(D

+

(J

I1 J1 I0

J3

M23 = 0

)

B2 A

( (

2

4 ),

11

2

+

= Nx0 µ2 + Ny0

2

)

)

M31 =

(J

I1 J1 I0

2

)

M32 = 0 M33 =

(

J12 I0

2)

2) 2)

177

J3

)

)

(z )2} dz

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha

=

22

=

(

23

=

µ 2(1

31

=

32

=

µ 2(1

=

(

33

=

˜13 = 1 2

=

2

1

BF A

µ2 2

1

2

1

+

)

(D

(

BF A

)

2

1

2

A¯ T (µ2 + 2) , 1

11

= ˜11

T

+

(

+

2

2

B2 2 A

2(1 + )

) (D

2

2mn(1

(

23 31

21 33

22 33

23 32

, ˜=

(

2)

nl

23 31

21 33

22 33

23 32

)+ ˜ ( 12

2)

B F A

)

2

B F A

)

B2 2 A

µ

8(1

21 32

22 31

22 33

23 32

B F A

)+

1 (A 2 2 (1 + )

+ 2 K + L)

F2 A

2

B F A

)+

1 (A 2 2 (1 + )

+ 2 K + L)

4 (B + F )( µ2 + 2) 2mn(1

A (µ4 + 2 µ2 2 + 4)

13

2

T¯mn, f3mn = (F¯ T + B¯ T )

1

, ˜12 =

)+ (

F2 A

2)

+ 2H + J

4µ (B + F )(µ2 +

2

F2 A

T¯mn, f2mn = (F¯ T + B¯ T )

2)

12

F2 A

B2 2 A

+ 2H + J

˜11 =

B2 2 A

+ 2H + J

(D + H ) (µ2 +

µ2

2

2)

)

B2 A

4B (µ4 + 2 µ2 2 + 4) 2mn(1

) (D

+ 2H + J

+

+

)

(D + H ) (µ2 +

2(1 + ) 2

(D

2

B2 A

+

µ2 + B¯ T 1

f1mn = T

(

µ

21

2)

)

21 32

22 31

22 33

23 32

)+ ˜

2)

T¯mn

1

,

)

13

dependent materials in thermal environment. J. Therm. Stress. 39, 854–873. https:// doi.org/10.1080/01495739.2016.1189772. Han, S.C., Park, W.T., Jung, W.Y., 2015. A four-variable refined plate theory for dynamic stability analysis of S-FGM plates based on physical neutral surface. Compos. Struct. 131, 1081–1089. https://doi.org/10.1016/j.compstruct.2015.06.025. Hao, Y.X., Chen, L.H., Zhang, W., Lei, J.G., 2008. Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate. J. Sound Vib. https://doi.org/10.1016/ j.jsv.2007.11.033. Huang, X.-L., Shen, H.-S., 2004. Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. Int. J. Solids Struct. 41, 2403–2427. https:// doi.org/10.1016/j.ijsolstr.2003.11.012. Joshan, Y.S., Grover, N., Singh, B.N., 2017. A new non-polynomial four variable shear deformation theory in axiomatic formulation for hygro-thermo-mechanical analysis of laminated composite plates. Compos. Struct. 182, 685–693. https://doi.org/10. 1016/j.compstruct.2017.09.029. Jung, W.Y., Park, W.T., Han, S.C., 2014. Bending and vibration analysis of S-FGM microplates embedded in Pasternak elastic medium using the modified couple stress theory. Int. J. Mech. Sci. 87, 150–162. https://doi.org/10.1016/j.ijmecsci.2014.05. 025. Jung, W.-Y., Han, S.-C., Park, W.-T., 2016. Four-variable refined plate theory for forcedvibration analysis of sigmoid functionally graded plates on elastic foundation. Int. J. Mech. Sci. 111–112, 73–87. https://doi.org/10.1016/j.ijmecsci.2016.03.001. Lee, W.-H., Han, S.-C., Park, W.-T., 2015. A refined higher order shear and normal deformation theory for E-, P-, and S-FGM plates on Pasternak elastic foundation. Compos. Struct. 122, 330–342. https://doi.org/10.1016/j.compstruct.2014.11.047. Li, Q., Iu, V.P., Kou, K.P., 2008. Three-dimensional vibration analysis of functionally graded material sandwich plates. J. Sound Vib. https://doi.org/10.1016/j.jsv.2007. 09.018. Merdaci, S., Tounsi, A., Houari, M.S.A., Mechab, I., Hebali, H., Benyoucef, S., 2011. Two new refined shear displacement models for functionally graded sandwich plates. Arch. Appl. Mech. https://doi.org/10.1007/s00419-010-0497-5. Meziane, M.A.A., Abdelaziz, H.H., Tounsi, A., 2014. An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J. Sandw. Struct. Mater. 16, 293–318. https://doi.org/ 10.1177/1099636214526852. Mortensen, A., Suresh, S., 1995. Functionally graded metals and metal-ceramic composites .1. Processing. Int. Mater. Rev. 40, 239–265. https://doi.org/10.1179/imr.1995. 40.6.239. Natarajan, S., Manickam, G., 2012. Bending and vibration of functionally graded material sandwich plates using an accurate theory. Finite Elem. Anal. Des. 57, 32–42. https:// doi.org/10.1016/j.finel.2012.03.006. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N., et al., 2013a. Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Compos. B Eng. 44, 657–674. https://doi.org/10.1016/j. compositesb.2012.01.089. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N., et al., 2013b. Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Compos. B Eng. 44, 657–674. https://doi.org/10.1016/j. compositesb.2012.01.089. Nguyen, D.D., 2018. Nonlinear thermo- electro-mechanical dynamic response of shear

References Abdelaziz, H.H., Atmane, H.A., Mechab, I., Boumia, L., Tounsi, A., Abbas, AB El, 2011. Static analysis of functionally graded sandwich plates using an efficient and simple refined theory. Chin. J. Aeronaut. 24, 434–448. https://doi.org/10.1016/S10009361(11)60051-4. Alijani, F., Bakhtiari-Nejad, F., Amabili, M., 2011. Nonlinear vibrations of FGM rectangular plates in thermal environments. Nonlinear Dynam. 66, 251–270. https://doi. org/10.1007/s11071-011-0049-8. Azizian, Z.G., Dawe, D.J., 1985. Geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method. Comput. Struct. https://doi.org/10.1016/00457949(85)90119-1. Beldjelili, Y., Tounsi, A., Mahmoud, S.R., 2016. Hygro-thermo-mechanical bending of SFGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory. Smart Struct. Syst. 18, 755–786. https://doi.org/10.12989/sss.2016.18. 4.755. Bennoun, M., Houari, M.S.A., Tounsi, A., 2016. A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates. Mech. Adv. Mater. Struct. 23, 423–431. https://doi.org/10.1080/15376494.2014.984088. Bessaim, A., Houari, M.S.A., Tounsi, A., Mahmoud, S., Bedia, E.A.A., 2013. A new higherorder shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets. J. Sandw. Struct. Mater. 15, 671–703. https://doi.org/10.1177/1099636213498888. Bourada, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A., 2012. A new four-variable refined plate theory for thermal buckling analysis of functionally graded sandwich plates. J. Sandw. Struct. Mater. https://doi.org/10.1177/1099636211426386. Dinh Duc, N., Hong Cong, P., 2018. Nonlinear thermo-mechanical dynamic analysis and vibration of higher order shear deformable piezoelectric functionally graded material sandwich plates resting on elastic foundations. J. Sandw. Struct. Mater. 20, 191–218. https://doi.org/10.1177/1099636216648488. Dinh Duc, N., Tuan, N.D., Tran, P., Quan, T.Q., 2017. Nonlinear dynamic response and vibration of imperfect shear deformable functionally graded plates subjected to blast and thermal loads. Mech. Adv. Mater. Struct. 24, 318–329. https://doi.org/10.1080/ 15376494.2016.1142024. Dogan, V., 2013. Nonlinear vibration of FGM plates under random excitation. Compos. Struct. 95, 366–374. https://doi.org/10.1016/j.compstruct.2012.07.024. Duc, N.D., Cong, P.H., 2015. Nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material plate with metal-ceramic-metal layers on elastic foundation. JVC J. Vib. Control 21, 637–646. https://doi.org/10.1177/ 1077546313489717. Duc, N.D., Tuan, N.D., Tran, P., Dao, N.T., Dat, N.T., 2015. Nonlinear dynamic analysis of Sigmoid functionally graded circular cylindrical shells on elastic foundations using the third order shear deformation theory in thermal environments. Int. J. Mech. Sci. 101–102, 338–348. https://doi.org/10.1016/j.ijmecsci.2015.08.018. Duc, N.D., Bich, D.H., Cong, P.H., 2016. Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations. J. Therm. Stress. 39, 278–297. https://doi.org/10.1080/01495739.2015.1125194. El Meiche, N., Tounsi, A., Ziane, N., Mechab, I., El, E.A., 2011. A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. Int. J. Mech. Sci. 53, 237–247. https://doi.org/10.1016/j.ijmecsci.2011.01.004. Fazzolari, F.A., 2016. Modal characteristics of P- and S-FGM plates with temperature-

178

European Journal of Mechanics / A Solids 76 (2019) 155–179

S.J. Singh and S.P. Harsha deformable piezoelectric sigmoid functionally graded sandwich circular cylindrical shells on elastic foundations. J. Sandw. Struct. Mater. 20, 351–378. https://doi.org/ 10.1177/1099636216653266. Nguyen, V.H., Nguyen, T.K., Thai, H.T., Vo, T.P., 2014. A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates. Compos. B Eng. 66, 233–246. https://doi.org/10.1016/j.compositesb.2014.05.012. Pandey, S., Pradyumna, S., 2015. Free vibration of functionally graded sandwich plates in thermal environment using a layerwise theory. Eur. J. Mech. A Solid. 51, 55–66. https://doi.org/10.1016/j.euromechsol.2014.12.001. Reddy, J., Chin, C., 1998. Thermomechanical analysis of functionally graded cylinders and plates. J. Therm. Stress. 37–41. Singh, S.J., Harsha, S.P., 2018. Nonlinear vibration analysis of sigmoid functionally graded sandwich plate with ceramic-FGM-metal layers. J. Vib. Eng. Technol. 18. https://doi.org/10.1007/s42417-018-0058-8. Singh, S.J., Harsha, S.P., 2019. Exact solution for free vibration and buckling of sandwich S-FGM plates on Pasternak elastic foundation with various boundary conditions. Int. J. Struct. Stab. Dyn. 19https://doi.org/10.1142/S0219455419500287. S0219455419500287. Sobhy, M., 2013. Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Compos. Struct. 99, 76–87. https://doi.org/10.1016/j.compstruct.2012.11.018. Thai, H.T., Nguyen, T.K., Vo, T.P., Lee, J., 2014. Analysis of functionally graded sandwich plates using a new first-order shear deformation theory. Eur. J. Mech. A Solid. 45, 211–225. https://doi.org/10.1016/j.euromechsol.2013.12.008.

Touloukian, Y.S., 1967. Thermophysical Properties of High Temperature Solid Materials. MacMillan, New York. Tounsi, A., Houari, M.S.A., Benyoucef, S., Adda Bedia, E.A., 2013. A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aero. Sci. Technol. 24, 209–220. https://doi.org/10.1016/j.ast.2011.11.009. Xiang, S., Kang, G wen, Yang, M sui, Zhao, Y., 2013. Natural frequencies of sandwich plate with functionally graded face and homogeneous core. Compos. Struct. https:// doi.org/10.1016/j.compstruct.2012.09.003. Xu, F., Zhang, X., Zhang, H., 2018. A review on functionally graded structures and materials for energy absorption. Eng. Struct. 171, 309–325. https://doi.org/10.1016/j. engstruct.2018.05.094. Zenkour, A.M., 2005. A comprehensive analysis of functionally graded sandwich plates: Part 2-Buckling and free vibration. Int. J. Solids Struct. 42, 5243–5258. https://doi. org/10.1016/j.ijsolstr.2005.02.016. Zenkour, A.M., Alghamdi, N.A., 2008. Thermoelastic bending analysis of functionally graded sandwich plates. J. Mater. Sci. 43, 2574–2589. https://doi.org/10.1007/ s10853-008-2476-6. Zenkour, A.M., Alghamdi, N.A., 2010. Bending analysis of functionally graded sandwich plates under the effect of mechanical and thermal loads. Mech. Adv. Mater. Struct. 17, 419–432. https://doi.org/10.1080/15376494.2010.483323. Zhang, W., Yang, J., Hao, Y., 2010. Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory. Nonlinear Dynam. https://doi. org/10.1007/s11071-009-9568-y.

179