Influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates

Composites Part B 163 (2019) 779–793

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Composites Part B journal homepage: www.elsevier.com/locate/compositesb

Influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates

T

Sanjay Singh Tomara, Mohammad Talhab,∗ a b

Aerospace Engineering Department, Indian Institute of Technology Kanpur, Uttar Pradesh, 208016, India School of Engineering, Indian Institute of Technology Mandi, Himachal Pradesh, 175005, India

ARTICLE INFO

ABSTRACT

Keywords: Functionally graded materials Material uncertainties Perturbation technique Monte Carlo simulation Finite element method

Present study aims to investigate the influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates. Reddy's higher order shear deformation theory has been employed to model the displacement field. Variational approach has been used to derive the governing differential equations. Effect of material uncertainties in the formulation have been incorporated using first order perturbation technique (FOPT). An efficient stochastic finite element formulation (SFEM) have been used for the calculation of first and second order statics of natural frequency and transverse deflection. Validation of the results have been performed with the help of available literature and separately developed Monte Carlo formulation (MCS) algorithm. A large number of examples have been solved to quantify the effect of uncertainties on the vibration and bending characteristics of functionally graded skew sandwich plates.

1. Introduction Laminated composites have found their wide application in various engineering field's. Basically laminated composite structures constitute of various isotropic layers stacked together by means of some bonding materials. The major drawback in this construction is the presence of stress concentrations at the interface of the various layers. Due to which laminated composites go through the delamination, cracking and other damages. This limitation of the laminated composites can be overcome with the help of functionally graded material (FGM's) [1]. FGM is an advance class of materials which have excellent thermo-mechanical properties which enables them to maintain their structural integrity in higher thermal loading conditions. Need of such material arises since the second world war when scientist started searching a material which can withstand higher temperature conditions [2]. They are usually composed of metal and ceramics. These constituent materials are graded in a fashion following various laws such as power law, sigmoid law, exponential law etc. The graded variation of FGM structures prevents them from delamination unlike the laminated composite materials [3]. The process of fabrication of FGM is a complex and challenging as a large number of design variables are involved which enables the smooth variation of material properties in the structures. In practical cases it is very difficult to have complete control on each step of

∗

manufacturing with full accuracy. Hence the inclusion of the deviation of material and geometric properties in the structures are inherent. This stochastic nature of material properties affects structural response of the system. Hence it becomes mandatory to involve such behaviour in the analysis. In recent years, this field have attracted the various researchers across the globe and as a result of which a large number of literature have been available in this field. Nakagiri et al. [4] studied vibration eigenvalue problem for FRP laminated plates with the help of second order perturbation method while considering various stacking sequence and layer thickness of the laminate. Oh and Librescu [5] studied the effect of uncertainties in the properties on the vibration and reliability of cantilever beam. In order to account the stochastic behaviour of uncertainties first and second order methods had been employed on the eigenvalues. Vanini and Mariani [6] investigated the effect of uncertainties on the vibration behaviour of composite plates resting on elastic foundation by solving the vibration problem in both deterministic and stochastic cases. Raj et al. [7] calculated the static response of composite plates with the help of MCS and finite element methodology. With the help of this analytical investigations authors predicted the behaviour of normally distributed parameters. Manohar and Ibrahim [8] reviewed the various work published in the area of vibration analysis of strictures having material uncertainties. Author covered various algorithm and methods such as finite element, boundary element method along with various

Corresponding author. E-mail address: [email protected] (M. Talha).

https://doi.org/10.1016/j.compositesb.2019.01.035 Received 23 April 2018; Received in revised form 2 January 2019; Accepted 3 January 2019 Available online 10 January 2019 1359-8368/ © 2019 Elsevier Ltd. All rights reserved.

Composites Part B 163 (2019) 779–793

S.S. Tomar, M. Talha

statistical techniques such as MCS and statistical energy analysis. Singh et al. [9] investigated the vibration response of laminated composite plates having material uncertainties. They used HSDT to account the effect of shear deformation and rotary inertia of the plates. FOPT had been used to model the stochastic nature of the system. The results obtained have been validated with MCS and compared with various plate models. Same group of authors performed another study [10] on the stochastic vibration of laminated composite cylindrical panel's. Considering uncertainties in the material properties. They employed the effect of transverse and rotary inertia in the calculation whereas FOPT had been employed to account the effect of uncertainties in the system. Onkar and Yadav [11] studied deflection response of laminated composite plates under random transverse loading conditions. They employed laminated plate theory along with von-karman assumptions for the formulation. FOPT had been used to calculate the transverse deflection of the problem whereas MCS had been used for the validation purpose. Same group of authors performed two more studies on laminated composites. Firstly they [12] calculated the non-linear vibration response of laminated composite plates. Later they [13] calculated the forced vibration response of laminated composite plates. They used Hamilton's principle to calculate the governing equation's for composite plates. Closed form solution along with FOPT had been used to calculate the variance of various responses. Yang et al. [14] performed stochastic bending response of FGM plates having uncertain material properties. They used HSDT in conjunction with FOPT to calculate the first and second order moment of deflection statics. Kitipornchai et al. [15] studied the vibration response of FG laminates having uncertain material properties in the thermal environment. They used HSDT for the formulation whereas, semi-analytical approach to solve the governing differential equations. FOPT had been used to calculate the variance of the eigen solution of the problem. Giunta and Carrera [16] proposed a Newton series approximation in the finite element environment analysis. In order to prove the improved computational efficiency with the presented formulation author solved the truss problem under static loading conditions. Tripathi et al. [17] analysed the stochastic vibration response of laminated composite conical shell structures considering the material properties as basic basic random variables. Effect of random material properties and random excitation on the reliability and vibration response of rotating beam had been studied by Hosseini and Khadem [18]. Gao [19] performed vibration analysis on the truss structure using random factor and interval factor methods. Author considered the frequency response and mode shapes to be random variable in analysis. Algebra synthesis method for random variables had been employed to derive the first and second order statics of the system as Rayleigh quotient. Lal et al. [20] investigated the stochastic bending behaviour of laminated composite plates resting on two parameter Pasternak foundation. They used a C 0 continuous finite element model in conjunction with FOPT to calculate the mean and variance of deflection statics. Same group of authors performed few more studies. Firstly they [21] calculated the vibraiton response of laminated composite plates resting on elatic foundation. Later they performed two nonlinear studies for calculation the nonlinear vibration [22] and nonlinear bending [23]of laminated composite plates resting on elastic. In both studies author used Von-Karman non-linear strain assumptions to account the non-linearity in the system. Plate was assumed to be under uniform pressure and thermal loading. Shaker et al. [24] used stochastic FEM methodology to calculate the vibration response of FGM plates. They used first and second order reliability approaches to calculate the variance of the eigenvalue solutions of the vibration problem. Singh et al. [25] presented a study on the non-linear bending response of laminated composite plates resting on two parameter elastic foundation. Authors employed HSDT in vonkarman sense for the formulation of the problem. Plate was assumed to be under the uniformly distributed loading conditions, MCS have been

employed for the validation purpose. The effect of material uncertainties on the deflection statics of sandwich composite plates have been investigated by Pandit et al. [26]. Authors developed an improved theory to calculate the structural response of sandwich plates. Theory was assumed to follow the stress continuity at the layer interfaces. Shahabian and Hosseini [27] studied the dynamic response of FG hollow cylinder having the material uncertainties. Authors employed Navier solution to calculate the displacement at each point. Whereas Newmark method with MCS had been employed to calculate mean and variance of the deflection statics. Same group of authors [28] performed one more study on the stress calculation on the FG hollow cylinder under the dynamic shock loading. The material properties of Al/ Al2 O3 were considered as basic random variables following Gaussian distribution. Chandrashekhar and Ganguli [29] used MCS with Latin hypercube sampling algorithm to perform the non-linear vibration analysis of laminated composite and sandwich composite plates. Pandit et al. [30] proposed an improved higher order zigzag theory and used it to calculate the vibration behaviour of soft core sandwich composite plates. Later same group of author's [31] performed another study using improved zigzag theory calculating the vibration response of sandwich composite plates. Theory relaxes the assumption of layerwise theory while considering the advantages of single later theories. They used FOPT to account the effect of material uncertainties. Lal and Singh [32] analysed the bending response of laminated composite plates under thermo-mechanical type of loading conditions. They employed a C 0 continuous finite element model along with FOPT to calculate deflection statics of the plates loaded under transverse loading conditions. Same group of authors [33] performed another study on the spherical shell panel considering material uncertainties. They used HSDT in Von-Karman sense for formulation. Shell was assumed to be under hygro-thermo-mechanical loading. Giunta et al. [34] performed free vibration analysis on the composite plates having material and geometric uncertainties. Author's used classical and two dimensional model via Carrera's unified formulation to formulate the eigenvalue problem. Monte Carlo method was used to incorporate the uncertainties. They prove that the frequency is not sensitive for the materialistic uncertainty as compared to the geometric uncertainty. Jagtap et al. [35] analysed the non-linear stochastic bending response of FGM plates in thermal environment. Authors used a C 0 continuous HSDT in conjunction with Von-Karman nonlinear assumptions. Thermal analysis had been done while considering temperature dependent and independent material properties. The noninear vibration analysis of composite plates laminated with piezoelectric layer having material uncertainty had been dome by Lal et al. [36]. Sepahvand et al. [37] used polynomial chaos free expansion to calculate the effect of material uncertainty in the elastic modulus on the free vibration response of orthotropic plates. Shegokar and Lal [38] performed a study on the deflection statics of the FGM beam beam bonded with piezoelectric layers having material uncertainties. Plate was assumed to be under thermo-electro-mechanical loading. They employed a direct iteration based FOPT to calculate the second order statics of the problem. Chang [39] investigated the non-linear dynamic behaviour of single walled carbon nano-tube using SFEM technique considering the effect of uncertainty in elastic moduli and mass density. Kumar et al. [40] studied the stochastic flexural response of laminated composite plates resting on elastic foundation having cubic nonlinearity. They modeled various material properties, lateral load, hygroscopic expansion coefficient as random variable. Dash and Singh [41] calculated the stochastic banding response of laminated composite plates. They used used a shear and normal deformation theory to account the effect of large deformation's. Green-Lagrange nonlinear assumptions have been considered to account the non-linearity in system. A C 0 continuous finite element formulation along with perturbation technique had been employed to calculate deflection statics of the composite plates. Talha and Singh [42] used an improved kinematics for the stochastic vibration analysis of FGM plates. Theory considers the 780

Composites Part B 163 (2019) 779–793

S.S. Tomar, M. Talha

Fig. 1. Geometry of plate in Cartesian coordinate system [59].

has been used to quantify the uncertainties. They proved that the sources of uncertainties effects the low velocity impact of the FGM plates. The present analysis aims to investigate stochastic bending and free vibration response of functionally graded skew sandwich plates. The formulations have been performed with Reddys higher order shear deformation theory (HSDT) by employing isoparametric C 0 continuous finite element. The governing equation's have been derived using variational approach. FOPT has been used to account the effect of material uncertainties in the functionally graded sandwich plate. Comparison and convergence studies have been performed to show the effectiveness and reliability of the present formulation. Separately developed Monte Carlo simulation methods for validation purposes. Various results have been shown to demonstrate the effect of skew angles, aspect ratio, thickness ratios, volume fraction index, and boundary conditions on the second order statics of defection and fundamental natural frequencies.

effect transverse and normal shear deformation's. In order to consider the effect of stochastic nature in the formulation material properties such as elastic modulus, volume fraction index of FGM plates had been considered as basic random variables. Xu et al. [43] employed random factor method to calculate stochastic dynamic characteristics of functionally graded beam and showed the characteristic effect of constituent volume fraction index on the dynamic characteristics distribution. Asnafi and Abedi [44] investigated the dynamic stability and bifurcation of FGM loads under stochastic random loading conditions. The properties of the plates were assumed to be varying in accordance with power law, exponential law and sigmoid law. They employed probability density function to evaluate the instability region and the response of the FGM plates had been solved using exact solution of Fokker Planck Kolmogorov equation. Chakraborty et al. [45] used PCFE (polynomial correlated function expansion) approach to perform the free vibration stochastic analysis on the composite plates. Garca-Macas et al. [46] studied the free vibration response of CNT reinforced FGM plates. The analysis had been performed with finite element methodology and probability theory was used to incorporate the uncertainties in the material properties. Mukhopadhyay et al. [47] calculated the effect of noise on stochastic free vibration behaviour composite laminated shallow shell structures. Authors used surrogate based methodology to quantify material uncertainties. Dey et al. [48] presented a Kriging model to analyse the free vibration response of composite shallow doubly curved shells. They used Mindlin's first order plate theory for formulation. Later same group of authors performed many studies which can be seen in Refs. [49–53]. Corradi et al. [54] performed a large number of experiments to calculate capacity and strength uncertainty of various types of wood beams. A special attention had been given to timber beams as they are commonly used for construction purpose. Tomar et al. [55] presented a critical review on the non-deterministic methodologies for the analysis of the composite structures. They performed a thorough survey on the various on the research work which had been performed in the last couple of centuries in the stochastic domain. Wu et al. [56] performed the static analysis on the Euler Bernoulli FGM beam while considering the uncertainties in various governing parameters such as geometry, material properties and loading's. Perturbation theory along with finite element methodology had been used to model the uncertainties and calculating second order deflection statics of the beam. Karsh et al. [57] studied the stochastic dynamic response of the twisted FGM plates using neural network based approach integrated with finite element formulation. They considered the uncertainties in the material and geometric properties of the plates and showed that the uncertainties significantly affect the dynamic behaviour of the functionally graded twisted plates. Same group of authors presented another study [58] on the FGM plates considering the low velocity response. They employed power law to calculate the material properties whereas the modified Hertzian contact law had been used to calculate the contact forces. Fuzzy logic based non-probabilistic method

2. Material properties and plate geometry definition Consider a rectangular sandwich FGM plate shown in Fig. 1 with sides a, b and height h respectively. Assuming that the mid-plane of plate coincides with the x-y plane. The gradation of FGM layer follows the power law distribution. The effective material properties are calculated using rule of mixtures (Vc + Vm = 1), (1)

Peff = Pc Vc + Pm Vm

Where, Peff represents the effective material properties such as young's modulus, Poisson's ratio, mass density and coefficient of elastic expansion, whereas Vc , Vm represents the volume fraction of the ceramic and metal, respectively. Plate having FGM core with homogeneous face sheet shown in Fig. 2 considered to follow given volume fractions [61],

Vc = 1, z

Vc =

( ) ,z z

h2

n

[h1, h/2] [h2, h1], hf = h1 + h2

hf

VC = 0, z

[ h /2, h2]

(2)

In order to consider the effect of thermal environment material properties are assumed to be the function of temperature.

P (t ) = P0 (P 1 T

1

+ 1 + P1 T + P2 T 2 + P3 T 3),

(3)

Where, P0, P 1, P1, P2, P3 represents coefficient of temperature and are unique to the type of materials. A linear temperature distribution has been employed across the thickness of the plate which is written as,

T (z ) = Tb + T

z 1 + h 2

(4)

Where, T = Tt Tb , Tt and Tb represents the temperature at top and bottom face of the functionally graded sandwich plate. 781

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Fig. 2. Cross section of Sandwich FGM plate with FGM core and homogeneous face sheets [60].

W0 x

x

W0 y

= 0,

=0

y

(7)

Final displacement vector at a node ‘i’ can be written as,

{ i} = {U0, V0, W0,

9

9

Ni { }i; i=1

Ni x i ;

y=

i=1

Ni yi

={}

(14)

i=1

3.1. Displacement field

x y

+ z2

0

x y

+ z3

0

U0 x = V0 + f1 (z ) y W0 0

+ f2 (z )

0 1

=

1 1

=

3 1

z 3,

U0 , x x

x

2 1 2 2

+ z2

1 3

V0 , 30 y

0 2

=

1 2

=

,

=

3C2 (

=

C2

(

x

=

C2

(

x

+ z3 0 0

3 1 3 2

;

3 3

0

(10)

x x

y

y

y

+ + +

=

1 3

,

y

+ x

=

y

W0 , 40 y

=

y

2 1

+

x

x ), x

x y

x

), +

3 1

= x

y

+

C2 x

y

( )

,

y

y

x

+

=

+

W0 , 50 x

3C2 (

y

y

),

y

+

=

V0 x y ),

+

U0 , y

2 2

3 1

1 2

(6)

0

0 0

Thermal strain vector can be represented as,

= 0 0 0

T (11)

Where, 1 and 2 represents the coefficient of thermal expansion in inplane directions respectively.

f2 (z ) = C3 C1 = 1, C2 = C3 = Where, f1 (z ) = C1 z C2 The complexities associated with the analysis of C1 continuous finite element is well known. To overcome these difficulty a C 0 continuitous displacement field have been prepared with panalty approch, due to this some artificial constants are imposed to the displacement field [62], z 3,

+z 0 0

1 1 1 2

Where

th

W0 x W0 y

=

yz xy

(5)

0

(8)

represents the linear strain vector and thermal strain

0 1 0 2 0 3 0 4 0 5

zx

Where, U,V and W represents inplane and transverse displacement at a point, U0, V0, W0 are the displacements corresponding to mid-plane. y and x represents the rotation of the normal to the mid-plane about y and x axis, x , y , x , y are the higher order terms of Taylor's series expansion. Value of the higher order terms are calculated with the help of zero transverse stress conditions at top and bottom surface of the plate. After applying these boundary condition following C1 continuous displacement field is obtained,

U V W

t

y

=

x y

T

(9)

x

The origin of Cartesian coordinate system is assumed to be lying on the edge of the sandwich plate shown in Fig. 1. Reddy's higher order shear deformation theory has been adopted for the mathematical formulation. As this theory includes the effect of transverse shear deformation and rotary inertia. According to Reddy's theory displacement field at a point can be written in terms of third order polynomial,

U0 = V0 + z W0

y}

{ th}

Where, ε and vector.

3. Mathematical formulation

U V W

x,

The strain displacement relationship for sandwich FGM plate can be represented as,

9

x=

y,

3.2. Strain displacement relationship

Fig. 3. Nide noded isoparametric element in local coordinate and global coordinate system. { }=

x,

4/3h2

3.3. Constitutive relationship The constitutive relation depicts the relationship among stress and strains of the sandwich FGM plate and is written as, 782

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S.S. Tomar, M. Talha

xx yy yz zx xy

Q11 Q12 0 0 0 Q12 Q22 0 0 0 0 Q44 0 0 { = 0 0 0 0 Q55 0 0 0 0 0 Q66

ne

t }or{ i}

l

e=1

= [Q¯ ij]{ i} (12)

E (z , T ) (z , T ) E (z , T ) Q11 = Q22 = , Q12 = , Q44 = Q55 = Q66 1 (z , T )2 1 (z , T ) 2 E (z , T ) = 2(1 + (z , T ))

N3 = N5 = N7 =

)(

1

2

), N2 = 4 (

2

+ ), N4 =

2 )( 2

), N6 =

+ )(

2 )( 2

+ ), N8 = 2 )(1

N9 = (1

2

+ )(

1 2 ( 4 1 2 ( + 2 1 2 ( 2 2)

2

)

2

+ )

)( )(1

2)

)(1

2)

(19)

e=1

(20)

D = [T ]Ti [Qij][T ]i (13)

4.1.2. Kinetic energy The kinetic energy of the plate is given by,

T=

1 2

{X}T {X} dv = v

1 2

NL A

k=1

hk + 1 hk

{X}T {X} dz dA

(21)

Where, ρ and X represents the mass density and the velocity vector for the plate respectively. The velocity vector can be written as,

In the present finite element formulation C 0 continuous 9 noded isoparametric Lagrangian element has been employed as shown in Fig. 3. Each node of the element contains 7 degree of freedom (DOF) per node and hence 63 DOF per element. The generalized displacement vector and element geometry at a point can be written as [63],Where, i represents the displacement vector at ith node and x i , yi are the Cartesian coordinates at ith node. Ni represents the interpolation functions which can be written as [64], 2

{ } e (T ) [K (e) ]{ } e

Where,

4. Finite element implementation

1 2 ( 4 1 (1 2 1 (1 2

ne

K (e) = [B]Ti [D][B ]i

Material properties in matrix [Qij ] are graded according to equations (1)–(3) respectively.

1

1 2

Where, ‘ne’ represent the number of elements present in the domain of the plate and K (e) be the element stiffness matrix for the plate can be written as,

Where , [Q¯ ] and [ ] represents the stress vector, reduced elastic coefficient matrix and strain vector respectively. The terms in the elastic coefficient matrix can be written as.

N1 = 4 (

U (e ) =

U=

(22)

{X} = [N¯ ]{ }

¯ ] is the function of the thickness coordinate can be written as, Where, [N 1 0 0 f1 (z ) 0 f2 (z ) 0 ¯] = 0 1 0 0 [N f1 (z ) 0 f1 (z ) 0 0 1 0 0 0 0

(23)

The kinetic energy of FGM laminate will be summation of each element. By applying the finite elements approximation the global kinetic energy expression can be written as,

T= (15)

1 2

ne

T

{ } (e) [M ](e) { } (e)

(24)

e=1

Where, m represents the elemental mass matrix for element ‘i’.

4.1. Energy calculation

4.2. Work done due to transverse load

4.1.1. Strain energy The strain energy for FGM plate element ‘i’ is given by Ref. [65],

The work potential generated due to distributed force acting on the top surface of the plate,

Ue =

1 2

v

{ }Ti { }i dv =

1 2

v

{ }Ti [Qij]{ }i dv

A

Where, ‘q’ represents distributed load vector acting on top surface of the plate.

Sandwich FGM plate consist of more than one number of layers, due to which for the calculation of strain energy material property matrix [Q] has to be calculated for each layer of sandwich plate and summed for all ‘NL’ nember of layers. New strain energy expression can be written as,

1 Ue = 2

NL A

k=1

hk + 1 hk

{ l}Ti [Qij]k { l}i dz dA

4.3. Skew boundary transformation Considering a skew plate shown in Fig. 4 whose boundaries are supported on the two adjacent edges, it is not necessary condition that the edges of the boundaries are parallel to the axis of the global coordinates. This fact leads us to define of the nodal displacements in terms of local coordinates and then transform it in terms of the global coordinates in order to define the boundary conditions. The transformation of the coordinates followed by the simple rules expressed as [66],

(17)

[T ] is the matrix of z extracted from strain terms, (18)

{ }i = [T ]{¯}i = [T ][B]i { }i

Where, [T] is the matrix made by extracting z terms from strain matrix and is given by,

1 0 T= 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

z 0 0 0 0

0 z 0 0 0

0 0 0 0 z

0 0 z2 0 0

0 0 0 z2 0

z3

0 0 z3 0 0 0 0 0 0

{ }T {q} dxdy

Wp =

(16)

i

= Tg

l i

(25)

Where, i , are the generalized and global displacement vectors at the respective i node. These can be defined as, l i

0 0 0 0 z3

{ i} = {U0, V0, W0,

x,

y,

x,

y}

T T

{ li} = U0l , V0l , W0l ,

The overall strain energy of the FGM sandwich plate will be summations of the stain energy of each individual element.

l x,

l y,

l x,

l y

The transformation matrix at node i can be written as, 783

(26)

Composites Part B 163 (2019) 779–793

S.S. Tomar, M. Talha

Fig. 4. Orientation of skew edges of plate.

Fig. 6. Comparison of COV ( 2 ) of Al considering Ec as random variable.

FGM

Al2 O3 sandwich plate with MCS

Table 1 Material properties of various constituent materials. Material

E (N /m2)

Properties ν

ρ(kg/m3)

Al Al2 O3 ZrO2

70 × 109

0.30

2707

151 × 109

0.30

3000

Ti

6Al

4V

380 × 109

0.30

105.7 × 109

0.298

Table 2 Variation of non-dimensional frequency parameter of Al sandwich plate with n = 1, a/h = 10. Mesh size

Table 3 Variation of non-dimensional central deflection of Al plate with n = 4.

3800

Mesh size

4429

FGM

Al2 O3

Frequency parameter 1:1:2

1:2:1

4 9 16 25 36 El Meiche et al. [68]

1.4024 1.3596 1.3485 1.3446 1.3429 1.3332

1.4966 1.4538 1.4429 1.4391 1.4375 1.4394

(%) Difference

0.7280

0.1304

FGM

Al2 O3 sandwich

Thickness ratio (a/h) 4

10

4 9 16 25 36 Neves et al. [69]

1.538 1.1968 1.0651 1.0656 1.0596 1.1108

1.1666 0.9146 0.8239 0.8225 0.8205 0.8700

(%) differ.

4.6095

5.6857

Fig. 7. Comparison of COV of transverse deflection of Ni / Al2 O3 FGM plate.

Fig. 5. Comparison of COV of square of natural frequency Al/ ZrO2 FGM plate. 784

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S.S. Tomar, M. Talha

elemental stiffness and mass matrices as,

KT = TTf K (e) Tf

(29)

MT = TTf M (e) Tf

(30)

Where, MT , KT represents the transformed global mass and stiffness matrices. 4.4. Governing equations In order derive the governing equations of plate variational principle have been adopted. The generalized governing differential equation for combined bending and vibration case can be written as, (31)

[MT ]{q¨} + [KT ]{q} = {F }

Assumption has been made that the system is vibrating in the principle mode, the eigenvalue problem for bending and vibration response of functionally graded skew sandwich plate can be written as, Fig. 8. Comparison of COV of transverse deflection of Al sandwich plate with MCS and FOPT.

c s 0 Tg = 0 0 0 0

c 0 0 0 0 0

s 0 0 1 0 0 0 0

0 0 0 c s 0 0

0 0 0

0 0 0 s 0 c 0 0 c 0 s

0 0 0 0 0 c

s

FGM

Al2 O3

Where, = 2 , ω represents the natural frequency of the functionally graded skew sandwich plate, q represents the global displacement vector, whereas F represents the global transverse load vector. 4.5. Perturbation approach (27)

In the present study in order to account the effect of material uncertainties in the vibration and bending response of functionally graded skew sandwich plates first order perturbation approach have been adopted. The basic material properties of skew sandwich plates are considered as basic input random variables. In the framework of perturbation technique, let following random field variable's have been assumed,

Where, c = cos( ) and s = sin( ), represent the skew angle for the plate. From the above transformation matrix it may be noted that for the various nodes which are not present on the skew edges transformation matrix have nonzero value only for the principle diagonal elements, which are equal to unity. Thus the transformation matrix for a complete element the element transformation matric can be written as,

Tf = Tg [I9]

(32)

[KT ]{q} = [MT ]{q} + {F }

(x , y ) = {

(28)

1 (x ,

y ),

2 (x ,

y ),

3 (x ,

y ), …………

m (x ,

y )}

(33)

Here elastic constants such as youngs modulus, poisons ratio and the volume fraction index are assumed as independent random variable. Using finite element considerations, variable's in the random field can be written as,

Where, I9 represents the identity matrix of order 9 × 9, the order of identity matrix will be same as that of number of nodes present in the element. Elemental transformation matrix will be used to transform the

Fig. 9. Variation of COV of square of natural frequency with thickness ratio(a/ h ) considering (a) Ec (b) Em as random variable. 785

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S.S. Tomar, M. Talha

Fig. 10. Variation of COV of square of natural frequency with volume fraction index(n) considering (a) Ec and Em (b) nn

m

=

Ni

i

and

m

as random variable.

m 0 0 i, j i, k Cov (

Var ( i ) =

(34)

i=1

c

r j,

r k)

(40)

j =1 k=1

As, material properties have been considered as random variables. Hence vectors and matrices in eq. (3.38) will be function of and will be random in nature.

where, Cov ( represents the coefficient of variance between and rk respectively.

[KT ] = [KT ( )] {qi } = {qi ( )} { } = { ( )}

Var (q) =

r j,

m

{qi} = { i} =

[KT0] + [KTr ] {qi0} + {qir } { i0} + { ir }

Exp (

0 0 i [MT ]{qi }

• First order equation

[KT0]{qir } + [KTr ]{qi0} = {F } +

[ ]=

+

r 0 i [MT ]{qi }

=

qir = [KTr ]i

=

m 0 r j = 1 i, d j m q0 r j = 1 i, j j m [KT0] rj j=1 i, j

r k)

r j)

(41)

represents the expectation matrix which can be

= [ ][ ][ ]

0 . . 0

0 ... 0 ... 0 . ... . , [ ] = . ... . 0 ...

1

12

21

. .

1 . .

k1

k2

... ... ... ... ...

1k 2k

. . 1

In this section, stochastic vibration and bending response of functionally graded skew sandwich plates various numerical results have been solved. This section have been divided into two major parts namely convergence and comparison study and the parametric studies. Stochastic results have been demonstrated in terms of coefficient of variance (COV=SD/mean) of square of natural frequencies and deflection statics of skew sandwich plates against COV of random variables (R.V.).

(37)

0 r i [MT ]{qi }

r j,

r j,

5. Results and discussions

(38)

Using first order Taylor series expansion, for small values of random part, terms [KTr ], ir , {qir } can be written as, r i

r j

r j

and, [ ] and [ ] are standard deviation and correlation coefficient matrices respectively.

(36)

• Zeroth order equation = {F } +

r j,

diag (qi, j )(qi, j )T Exp

Where, Exp ( calculated as,

Considering the variation in random part be very small as compared to mean value. Using the above equation and putting in equation (32) while collecting terms with same order and neglecting terms above first order following two equations can be derived,

[KT0]{qi0}

m

j =1 j=1

(35)

In FOPT random variable are expanded using the Taylor series expansion and upto first order terms of expansions are retained. Here, variables in the random field are expanded about mean values 0 of random variables, and zero mean random part r [67].

[KT ] =

r k)

5.1. Convergence and comparison studies In this section accuracy and applicability of the present formulation has been demonstrated through various examples. The current approach has been validated against the results available in the published literature and using Monte Carlo simulation (MCS) technique. This section aims to demostrate appropriate mesh size which can be used to obtain the results.

(39)

In the current case, results for vibration and bending case are calculated separately. The variance for eigenvalues of frequency and deflection can be written as, 786

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Fig. 11. Variation of COV of square of natural frequency with skew angle (ψ) considering (a) Ec (b) Em (c)

5.1.1. Vibration analysis Example 1: This example presents the validation study for frequency parameter ( ¯ ) of sandwich FGM plates simply supported (SSSS) at all edges have been validated against the results of El Meiche et al. [68]. Table 2 shows that the results obtained by the current formulation agrees well with the literature El Meiche et al. [68] at various volume

c

(d)

m

(e) Ec , Em (f) Ec ,Em ,

c , m ,n

as random variable.

fractions. Author's used a new hyperbolic shear deformation theory along with the Navier's closed form solution to calculate vibration response of the plate. The elastic modulus of metal ( Al ) and ceramic ( Al2 O3 ) have been considered to be 70GPa and 380Gpa , respectively. The material properties are also given in Table 1. The thickness ratio (a/ h ) has been taken as 10, frequency parameter ( ¯ ) used in study can 787

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Fig. 12. Variation of COV of square of natural frequency considering (a) all properties as R.V. (b) Various boundary conditions.

be written as:

¯ =

(b2 / h)

0 / E0

Example:2 This example presents the validation of deflection statics of Ni /Al2 O3 FGM plate having volume fraction index(n) and thickness ratio(a/ h ) to be 2 and 10 respectively. Elastic modulus of metal (Em ) is considered to be independent random variable. Fig. 7 shows that results obtained with current formulation agrees fairly well with the literature [14]. Yang et al. [14] used semi-analytical method based HSDT to calculate transverse deflection statics of FGM plate. Example:3 In this example comparative study have been performed between transverse deflection statics of Al FGM Al2 O3 sandwich FGM plate obtained with FOPT and MCS. Plate is assumed to be under uniformly distributed loading and simply supported boundary conditions. Elastic modulus of ceramic(Ec ) has been considered to be random variable. The volume fraction index(n) and thickness ratio(a/ h ) to be 2 and 10 respectively. Fig. 8 shows that results obtained with FOPT agrees with MCS and thus current formulation can be used to produce results for sandwich FGM plate.

(42)

where, ω, b and h are natural frequency, width and thickness of plate. 0 and E0 are constant terms in frequency parameters having values 1Kg / m3 and 1GPa , respectively. Example:2 In this example validation of coefficient of variation (COV) of square of natural frequency ( 2 ) of square Al/ ZrO2 FGM plate with Shaker et al. [24] has been performed. The volume fraction index (n) and thickness ratio (a/ h ) in the study had been considered as 2 and 10, respectively. Plate is assumed to be simply supported at all edges. The elastic modulus of metal and ceramic have been considered to be 70 GPa and 151 Gpa, respectively. Shaker et al. [24] employed first and second order reliability method whereas in present study FOPT has been used to obtain the second order statics of vibration response of the FGM plates. Fig. 5 shows that results obtained from the present formulation agrees well while considering elastic modulus (Ec and Em ) as random variables (R. V .). Example:3 In this example coefficient of variation (COV) of square of natural frequency ( 2 ) of simply supported square Al FGM Al2 O3 sandwich FGM plate is validated with independently developed Monte Carlo simulation (MCS) and FOPT approach. Elastic modulus of ceramic (Ec ) has been considered as random variable (R.V.). Thickness ratio (a/ h ) and volume fraction index (n)of simply supported plates has been considered as 10 and 2, respectively. The elastic modulus of metal and ceramic have been taken to be 70 GPa and 380 Gpa, respectively. The results obtained MCS and FOPT matches well as shown in Fig. 6. Since, the MCS is computationally expensive, therefore FOPT approach has been used to obtain the results for stochastic vibration response of sandwich FGM plates.

5.2. Parametric studies Objective of the parametric studies is to represent the effect of dispersion of various material properties and random variables on the COV of square of natural frequency( 2 ) and transverse deflection(w) on Al FGM Al2 O3 skew sandwich plate. The material properties in the study have been assumed to be varying from 0% to 15%. The thickness ratio of each layer is consider to be in 1:1:2 ratio moving from top to bottom layer. The material properties used in the analysis are as follows:

• Aluminium(Al): Elastic modulus (E ) = 70 GPa, Mass density ( ) = 2707 Kg / m , Poisson's ratio ( ) = 0.3 • Aluminium Oxide ( Al O ): Elastic modulus (E ) = 380 GPa, Mass m

m

3

m

2

3

c

density ( c ) = 3800 Kg / m3 , Poisson's ratio ( c ) = 0.3

5.1.2. Bending analysis Example:1 In this example the non-dimensional central deflection of Al FGM Al2 O3 sandwich FGM plate has been validated with Neves et al. [69]. Validation has been performed while considering different values of thickness ratios(a/ b ) whereas, volume fraction index (n) has been considered to be 4. Neves et al. [69] used quasi 3D HSDT while incorporating collocation method having radial basis function. Table 3 shows comparison of results from both the methodologies. It can be seen that at 6 × 6 mesh the results obtained with current formulation agrees well with the published literature [69].

5.2.1. Vibration analysis Example:1 This example presents the variation in COV of square of natural frequency ( 2 ) of Al FGM Al2 O3 skew sandwich plate with thickness ratio (a/ h ) by considering elastic modulus (Ec and Em ) as random variables. Plate has been considered to be simply supported at all edges. The volume fraction index (n) and skew angle (ψ) of square sandwich plate is taken to be 2 and 100 , respectively. It is observed from Fig. 9 that the values of COV of square of natural frequency ( 2 ) 788

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Fig. 13. Variation of COV of transverse displacement with thickness ratio (a/ h ) considering (a) Ec (b) Em (c)

increases with increase in thickness ratio. This is more predominant in case of Ec as compared to Em . This is due to the fact that the value of elastic modulus of ceramic is higher than that of metal. Example:2 In this example the variation of COV of square of natural frequency ( 2 ) of simply supported Al FGM Al2 O3 skew sandwich plate with volume fraction index (n) considering elastic moduli (Ec and

c

(d)

m

(e) n (f) Ec ,Em as random variable.

Em ) and poisson's ratio's ( c and m ) have been demonstrated. The thickness ratio (a/ h ) and skew angle (ψ) of skew sandwich plat are considered to be 10 and 100 , respectively. Fig. 10 shows that COV of 2 varies in cases where Em and m are varying. This is mainly due to the fact that on increasing the volume fraction index (n) of the plate, metal fraction in FGM layer increases and hence the variation in the COV can 789

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Fig. 14. Variation of COV of transverse displacement with thickness ratio (a/ h ) considering (a) Ec (b) Em (c)

be observed. Example:3 In this example various results for COV of square of natural frequency ( 2 ) of Al FGM Al2 O3 skew sandwich FGM plates with skew angle (ψ) has been demonstrated while varying all the material properties (Ec , Em , c , m , n) one by one. The thickness ratio (a/ h ) and volume fraction index (n) of the simply supported plates have been considered to be 2 and 10, respectively. It can be observed from Fig. 11 that in all the cases the value of COV of 2 increases with increase of skew angle (ψ). This is due to the increase in mean natural frequency of the system with skew angle. The numeric values of COV, 2 is more in case of elastic moduli (E and E ) is more than poissons c m ratios ( c and m ) as the numeric mean values of elastic moduli are more than poissons ratios. Due to which small deviation in elastic modulus largely influences the dispersion behaviour of natural frequency of system. Example:4 In this examples, two special cases have been demonstrated. Firstly by considering the effect of increase of random material properties one by one on COV of square of natural frequency ( 2 ) later by considering various boundary conditions. It can be observed from

c

(d)

m

as random variable.

Fig. 12a that with increase in number of independent random variables (R.V.) COV of natural frequency ( 2 ) increases. This shows that the effect of randomness in material properties have a significant effect on sensitivity of natural frequency of plate. It is also observed from Fig. 12b that the value of COV of natural frequency ( 2 ) is comes out to be maximum in case of SFSF boundary conditions. This can be understood as the this condition offers least constrained condition among various boundary conditions used, where as the value is minimum in case of CCCC boundary conditions as it offers highly constrained condition for the plate. 5.2.2. Bending analysis In this section various results have been shown while considering square Al FGM Al2 O3 skew sandwich plates under simply supported boundary condition and with uniform loading conditions of 100 MPa. Example:1 This examples presents the dispersion of transverse displacement (w) of square Al FGM Al2 O3 skew sandwich plate with thickness ratio (a/ h ) while considering all properties (Ec , Em , c , 790

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Fig. 15. Variation of COV of transverse displacement with volume fraction index (n) considering (a) Ec (b) Em (c)

c

(d)

m

as random variable.

Fig. 16. Variation of COV of square of natural frequency considering all properties as R.V. m ,n) as random variables shown in Fig. 13. Plates are assumed to be under simply supported boundary condition. The volume fraction index (n) and skew angle (ψ) are considered as 2 and 100 , respectively. It has been observed that COV (S. D. / Mean ) of transverse displacement (w) decreases with the increase in thickness ratio (a/ h ) while considering Ec and c as random variable. While contrary COV of transverse displacement (w) decreases with increase in Em , n and m . In general it can be observed that COV of transverse displacement (w) is most sensitive to ceramic elastic modulus (Ec ) and metal poisson's ratio ( m ). Example:2 In this example the effect of variation of volume fraction index (n) on COV (S. D. / Mean ) of transverse displacement (w) have been observed considering Ec , Em , c , m as independent random variables. The thickness ratio (a/ h ) and skew angle (ψ) of the plate have been considered to be 10 and 100 , respectively. The variation has been observed at n = 1, 2, 5 and 10, respectively. It has been observed from Fig. 14 that dispersion of COV of transverse displacement (w) increases

in case of Em and m on the contrary COV decreases with increase in volume fraction index (n) while considering Ec and c as random variables. This is mainly due to the fact that with the increase in volume fraction index (n) metal volume fraction in FGM layer of sandwich plate increases. Due to which the sensitivity of transverse deflection (w) can be easily observed. Example:3 This example presents the effect of dispersion of volume fraction index(n) on COV of transverse displacement(w) of Al FGM Al2 O3 skew sandwich plate considering more than one material properties as random variable. The Thickness ratio and skew angle of the plate have been considering to be 10 and 100 , respectively. Plate is assumed to be under simply supported boundary conditions. From Fig. 15 it can be observed that with increase in volume fraction index(n) COV of transverse displacement(w) increases with an exception at n = 1. It can be understood as elastic modulus of ceramic(Ec ) and metal(Em ) being considered to be random. With results in increase 791

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in COV of transverse displacement(w) due to (Em ) and decrease sue to (Ec ). Hence due to combined effect of both random variable such interesting result has been obtained. Example:4 This example shows the effect of increase of random variable on the dispersion of transverse displacement (w) of Al FGM Al2 O3 skew sandwich plate. Volume fraction index (n), thickness ratio (a/ h ) and skew angle (ψ) of plate have been considered to be 2, 10 and 100 , respectively. It has been observed from Fig. 16 that COV of transverse displacement (w) increases with increase in number of independent random variable. This shows that all the material properties leads to the increase in the sensitivity of the transverse displacement (w). This example shows the effect of increase of random variable on the dispersion of transverse displacement(w) of Al FGM Al2 O3 skew sandwich plate. Volume fraction index(n), thickness ratio(a/ h ) and skew angle(ψ) of plate have been considered to be 2,10 and 100 , respectively. It can be observed from Fig. 16 that COV of transverse displacement(w) increases with increase in number of independent random variable. This shows that all the material properties leads to the increase in the sensitivity of the transverse displacement(w).

References [1] Zidi M, Tounsi A, Houari MSA, Adda Bedia EA, Anwar Bég O. Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory. Aero Sci Technol 2014;34(1):24–34. https://doi.org/10.1016/j.ast. 2014.02.001. [2] Koizumi M. FGM activities in Japan. Compos B Eng 1997;28(1–2):1–4. https://doi. org/10.1016/S1359-8368(96)00016-9. [3] Gupta A, Talha M. Recent development in modeling and analysis of functionally graded materials and structures. Prog Aero Sci 2015;79:1–14. https://doi.org/10. 1016/j.paerosci.2015.07.001. [4] Nakagiri S, Takabatake H, Tani S. Uncertain eigenvalue analysis of composite laminated plates by the stochastic finite element method. J. Eng. Ind. 1987;109(1):9. https://doi.org/10.1115/1.3187096. [5] Oh D, Librescu L. Free vibration and reliability of composite cantilevers featuring uncertain properties. Reliab Eng Syst Saf 1997;56(3):265–72. https://doi.org/10. 1016/S0951-8320(96)00038-5. [6] Venini P, Mariani C. Computational structures technology free vibrations of uncertain composite plates via stochastic Rayleigh-Ritz approach. Comput Struct 1997;64(1):407–23https://doi.org/10.1016/S0045-7949(96)00161-7. [7] Navaneetha Raj B, Iyengar N, Yadav D. Response of composite plates with random material properties using FEM and Monte Carlo simulation. Adv Compos Mater 1998;7(3):219–37. https://doi.org/10.1163/156855198X00165. [8] Manohar CS, Ibrahim RA. Progress in structural dynamics with stochastic parameter variations: 1987-1998. Appl Mech Rev 1999;52(5):177. https://doi.org/10.1115/1. 3098933. [9] Singh BN, Yadav D, Iyengar NGR. Natural frequencies of composite plates with random material properties using higher-order shear deformation theory. Int J Mech Sci 2001;43(10):2193–214. https://doi.org/10.1016/S0020-7403(01) 00046-7. [10] Singh BN, Yadav D, Iyengar NGR. Free vibration of composite cylindrical panels with random material properties. 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Random vibration of the functionally graded laminates in thermal environments. Comput Methods Appl Mech Eng 2006;195(9–12):1075–95. https://doi.org/10.1016/j.cma.2005.01.016. [16] Giunta G, Carrera E. Stochastic static analyses of fe models by means of newton's series expansions. III European Conference on Computational Mechanics. Dordrecht: Springer Netherlands; 2006. p. 293. ISBN 978-1-4020-5370-2;. [17] Tripathi V, Singh B, Shukla K. Free vibration of laminated composite conical shells with random material properties. Compos Struct 2007;81(1):96–104. https://doi. org/10.1016/j.compstruct.2006.08.002. [18] Hosseini S, Khadem S. Vibration and reliability of a rotating beam with random properties under random excitation. Int J Mech Sci 2007;49(12):1377–88. https:// doi.org/10.1016/j.ijmecsci.2007.04.008. [19] Gao W. Natural frequency and mode shape analysis of structures with uncertainty. Mech Syst Signal Process 2007;21(1):24–39. https://doi.org/10.1016/j.ymssp. 2006.05.007. 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6. Conclusion Stochastic vibration and bending response of functionally graded skew sandwich plates have been studied while considering the uncertainty in material properties. Reddy's HSDT has been used to model the displacement field. Variational principle has been used to derive governing differential equations. Finite element methodology in conjunction with FOPT have been used to calculate the first and second order statics of natural frequency and transverse displacement of skew sandwich plate. Convergence and validation studies have been performed to show the reliability of present formulation. Various numerical results have been shown to observe the effect of various random variables and system parameters on the vibration and deflection statics of functionally graded skew sandwich plates. The list of conclusions that can be drawn from current study are as follows:

• COV of natural frequency( ) and transverse displacement(w) varies linearly with COV of random material properties. • Effect of increase of thickness ratio(a/h) on COV of natural fre2

• • • • • •

quency( 2 ) has been comes out to be more while considering Ec as random variable compared to Em . Effect of increase of volume fraction index(n) on COV of natural frequency( 2 ) have been comes out to be maximum while considering Em and m as random variable whereas effect is minimum for Ec and c , respectively. The dispersion of elastic modulus of ceramic(Ec ) have maximum effect on the sensitivity of natural frequency of skew sandwich FGM plate on contrary it found to be minimum in case of m . COV of natural frequency( 2 ) increases with increase in number of independent random variables considering other parameter to be constant. As plate thickness ratio(a/h) the sensitivity of transverse deflection (w) increases while considering Em and m as R.V. whereas, on contrary sensitivity decreases while considering Em , m and n as random variable. Sensitivity of transverse deflection(w) increases while considering Em and m as random variable whereas, on contrary it decreases while considering Ec as random variable. COV of transverse deflection(w) increases with the increase in number of independent random variables.

Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.compositesb.2019.01.035. 792

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