Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations

Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations tài liệu, giáo án, bài giảng ,...

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Journal of Thermal Stresses

ISSN: 0149-5739 (Print) 1521-074X (Online) Journal homepage: http://www.tandfonline.com/loi/uths20

Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations Nguyen Dinh Duc, Dao Huy Bich & Pham Hong Cong

To cite this article: Nguyen Dinh Duc, Dao Huy Bich & Pham Hong Cong (2016) Nonlinear

thermal dynamic response of shear deformable FGM plates on elastic foundations, Journal ofThermal Stresses, 39:3, 278-297

To link to this article: http://dx.doi.org/10.1080/01495739.2015.1125194

Published online: 15 Mar 2016

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2016, VOL 39, NO 3, 278–297

http://dx.doi.org/10.1080/01495739.2015.1125194

Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations

Nguyen Dinh Duca, Dao Huy Bicha, and Pham Hong Conga , b

aAdvanced Materials and Structures Laboratory, University of Engineering and Technology, Vietnam National University,Hanoi, Vietnam;bCenter for Informatics and Computing, Vietnam Academy of Sciences and Technology, Hanoi, Vietnam

ABSTRACT

This paper investigates the nonlinear dynamic response of thick functionally

graded materials (FGM) plates using the third-order shear deformation plate

theory and stress function The FGM plate is assumed to rest on elastic

foun-dations and subjected to thermal and damping loads Numerical results for

dynamic response of the FGM plate are obtained by Runge–Kutta method The

results show the influences of geometrical parameters, the material properties,

the elastic foundations, and thermal loads on the nonlinear dynamic response

of FGM plates

ARTICLE HISTORY

Received 2 April 2015 Accepted 17 May 2015

KEYWORDS

FGM plate; nonlinear dynamic response; stress function; the third-order shear deformation plate theory; thermal loads

Introduction

Functionally graded materials (FGMs) are microscopically inhomogeneous composite materials inwhich the mechanical and thermal properties vary smoothly and continuously from one surface tothe other Typically, these materials are made from a mixture of metal and ceramic or a combination

of different metals by gradually varying the volume fraction of the constituent metals The properties

of FGM plates and shells are assumed to vary through the thickness of the structures Due to thehigh temperature resistance, FGMs have many practical applications such as reactor vessels, aircrafts,space vehicles, defense industries, and other engineering structures As a result, in recent years, manyinvestigations have been carried out on the dynamic and vibration of FGM plates and shells

The vibration of functionally graded cylindrical shells has been investigated by Lam and Reddy [1].Lam and Hua considered the influence of boundary conditions on the frequency characteristics of arotating truncated circular conical shell [2] In [3], Pradhan et al studied vibration characteristics ofFGM cylindrical shells under various boundary conditions Yang and Shen [4] published the nonlinearanalysis of FGM plates under transverse and in-plane loads Zhao et al [5] studied the free vibration

of two-side simply supported laminated cylindrical panel through the mesh-free kp-Ritz method.About vibration of FGM plates, Vel and Batra [6] gave a three-dimensional exact solution for thevibration of FGM rectangular plates Ferreira et al [7] received natural frequencies of FGM plates bymeshless method Woo et al [8] investigated the nonlinear free vibration behavior of functionally gradedplates Kadoli and Ganesan [9] studied the buckling and free vibration analysis of functionally gradedcylindrical shells subjected to a temperature-specified boundary condition Wu et al [10] published theirresults of nonlinear static and dynamic analysis of functionally graded plates Natural frequencies andbuckling stresses of FGM plates were analyzed by Matsunaga [11] using two-dimensional higher-orderdeformation theory Shariyat obtained the dynamic thermal buckling of suddenly heated temperature-dependent FGM cylindrical shells under combined axial compression [12] and external pressure anddynamic buckling of suddenly loaded imperfect hybrid FGM cylindrical with temperature-dependentmaterial properties under thermoelectromechanical loads [13] Zhao et al [14] studied free vibration

CONTACTNguyen Dinh Duc ducnd@vnu.edu.vn Advanced Materials and Structures Laboratory, University of Engineering and Technology, Vietnam National University, 144 - Xuan Thuy Street, Hanoi, Vietnam.

analysis of functionally graded plates using the element-free kp-Ritz method Bich and Nguyen [15]investigated nonlinear vibration of functionally graded circular cylindrical shells based on improvedDonnel equations using classical shell theory Recently, Duc [16] published a valuable book “NonlinearStatic and Dynamic Stability of Functionally Graded Plates and Shells”, in which the results aboutnonlinear vibration of shear deformable FGM plates and shells are presented Mohammad and Singh[17] studied static response and free vibration of unsymmetrical FGM plates using first-order sheardeformation theory with finite element method Duc and Cong [18] studied nonlinear dynamic response

of imperfect symmetric thin S-FGM plate with metal–ceramic–metal layers on elastic foundation [18].Duc [19] has recently investigated the nonlinear dynamic response of imperfect eccentrically stiffenedFGM double curved shallow shells on elastic foundation

Up to date, dynamic analysis of FGM plates and shells using the higher-order shear deformationtheory has received great attention of the researchers Ungbhakorn and Wattanasakulpong [20] studiedthermoelastic vibration analysis of third-order shear deformable functionally graded plates with dis-tributed patch mass under thermal environment, the solutions are obtained by energy method Talhaand Singh [21] studied static response and free vibration analysis of FGM plates using higher-order sheardeformation theory with finite element models Huang and Shen [22] studied nonlinear vibration anddynamic response of FGM plates in thermal environments but volume fraction follows a simple powerlaw for unsymmetrical FGM plate Swaminathan and Ragounadin [23] studied analytical solutionsusing a higher-order refined theory for the static analysis of antisymmetric angle-ply composite andsanwich plates Qian et al [24] studied static and dynamic deformations of thick functionally gradedelastic plates using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin method In all the metioned publications [20–24], the authors used finite element method andthe displacement functions

With some other publications [15,18,19], the authors have used the stress function and Volmir’sassumption with analytical approach to investigate the vibration and dynamic responses of FGM platesand shells applying classical plate and shell theories for thin structures For thick structures, we have

to use the third-order shear deformation theory, but then Volmir’s assumption is useless There hasnot been any publication using the third-order shear deformation plate theory and stress function tostudy vibration and dynamic response for FGM plates Therefore, this paper investigates the nonlinearthermal dynamic response and nonlinear vibration of thick FGM plates using the third-order sheardeformation plate theory and stress function (in two cases: uniform temperature rises and nonlineartemperature through the thickness of the plate) Numerical results for dynamic response of the FGMplate are obtained by Runge–Kutta method

Theoretical formulation

Consider a rectangular FGM plate on elastic foundations The plate is referred to a Cartesian coordinate

system x, y, z, where xy is the midplane of the plate and z is the thickness coordinator, −h/2 ≤ z ≤ h/2 The length, width, and total thickness of the plate are a, b, and h, respectively (Figure 1)

Figure 1. Geometry of the FGM plate on elastic foundations.

By applying a simple power law distribution (P-FGM), the volume fractions of metal and ceramic,

V m and V c, are assumed as [25]

Vc(z) =  2z + h

2h

N

where the volume fraction index N is a non-negative number that defines the material distribution and

can be chosen to optimize the structural response

The effective properties Peffof the FGMs are determined by the modified mixed rules as follows [25]:

and ceramic constituents, respectively

From Eq (1) and Eq (2), the effective properties of the FGM plate can be written as follows:

[E(z), α(z), ρ(z), K (z)] = [Em, αm, ρm, Km] + [Ecm, αcm, ρcm, Kcm] 2z + h

2h

N

(3)where

Ecm=Ec−Em, αcm=αc−αm, ρcm=ρc−ρm, Kcm=Kc−Km (4)

and the Poisson ratio v(z) is assumed to be constant v(z) = v.

Suppose that the FGM plate is subjected to a transverse load and compressive axial loads In this study,the Reddy’s third-order shear deformation theory is used to obtain the motion, compatibility equations

At the same time, the stress function is applied for determining the nonlinear dynamic response andvibration of the FGM plate

The strain–displacement relations taking into account the Von Karman nonlinear terms and thethird-order shear deformation plate theory are [26–28]

in which c1 = 4/3h2, εx, εyare normal strains, γxy is the in-plane shear strain, and γxz, γyzare the

transverse shear deformations Also u, v, ω are the displacement components along the x, y, z directions,

respectively, and φx, φy are the slope rotations in the (x, z) and (y, z) planes, respectively.

The strains are related in the compatibility equation [27,28]

Z h/2

−h/2

(1, z, z3)E(z)α(z)1T(z)dz

and specific expressions of coefficients E i(i = 1/5, 7) are given in Appendix.

For using late, the reverse relations are obtained from Eq (10)

According to higher-order shear deformation theory, the equations of motion are [26]

and specific expressions of coefficients I i(i = 0 ÷ 4, 6) are given in Appendix, and k1 is Winkler

foundation modulus, k2 is the shear layer foundation stiffness of Pasternak model, q is an external

pressure uniformly distributed on the surface of the plate, ε is damping coefficient

The stress function f (x, y, t) is introduced as

By setting Eq (12) and Eq (15) into the deformation compatibility Eq (7), we obtain

and the linear operators Lij i = 1 − 3, j = 1 − 3
and the nonlinear operator P are given in Appendix.

The system of four Eqs (18) and (19) combined with boundary conditions and initial conditionscould be used for nonlinear dynamical analysis of FGM plates using the higher-order shear deformationtheory in the next section

Nonlinear dynamic analysis

Four edges of the plate are simply supported and immovable (IM) In this case, boundary conditions are[27,28]

w = u = φ y=M x =P x=0, N x =N x0 at x = 0, a

(20)

w = v = φ x =M y=P x=0, N y=N y0 at y = 0, b

in which N x0 , N y0are the forces are the jets when the edges are IM in the plane of the plate

The approximate solutions of the system of Eq (18) and Eq (19) satisfying the boundary condition[Eq (20)] can be written as [26]:

By substituting displacement functions [Eq (21)] into compatibility Eq (18), we define the stressfunction as

f x, y, t
=A1(t) cos 2αx + A2(t) cos 2βy +1

0 −c1J4 Specific expressions of coefficients

lij i = 1 ÷ 3, j = 1 ÷ 3
, and n1, n2are given in Appendix

Equation (23) is used to analyze the nonlinear dynamic response of thick FGM plates on the elasticfoundation applying the higher-order shear deformation theory in thermal environment

Consider the FGM plate with all edges which are simply supported and IM (all edges IM) under

thermal load The condition expressing the immovability on the edges, u = 0 (on x = 0, a) and v = 0 (on y = 0, b), is satisfied in an average sense as [27,28]

 ∂w

∂x

2+8a

 ∂w

∂y

2+8a

By replacing Eqs (26a) and (26b) into the equations of motion [Eq (23)], we have



l11+



mπ a

2+

nπ b

and specific expressions of coefficients l 1i(i = 4 − 7) are given in Appendix The effects of temperature

in Eq (27) appear in 8a 8a are identified upon the uniform increasing temperature or nonlineartemperature transferring through the thickness of the plate

Taking linear parts of the set of Eq (27) and placing q = 0, the natural frequencies (ω) of the plate

can be determined directly by solving determinant

l11+



mπ a

2+ nπ b
2

Solving Eq (28) yields three frequencies of the FGM plates, and the smallest one is being considered

Uniform temperature rises

The plate is placed in the environment which temperature is steadily increased from the beginning value

T i to the ending value T f , the temperature difference 1T = T f −T iis a constant

Where 8ais determined from Eq (11) and

Through the thickness temperature gradient

In many applications of FGM plates, the temperature distribution in the plate is uneven Usually muchhigher temperature in rich ceramic surface than the surface of the rich metal sheet of FGM plate Inthis case, the temperature through thickness of plate is described by the Fourier temperature equation

By switching Eq (31) into Eq (11), we obtain 8a

and specific expression of coefficients H are given in Appendix.

Numerical results and discussion

Consider a FGM plates acted on by an uniformly distributed transverse load q = Q sin t (Q is the

amplitude of uniformly excited load,  is the frequency of the load) The fourth-order Runge–Kuttamethod is used to solve Eq (27) in which 8ais identified through Eq (29) in the case of the uniformincreasing temperature and in the case of the nonlinear temperature transferring through the thickness

of the structure, 8ais taken from Eq (32) To illustrate the present approach, we consider a ceramic–metal FGM plate that consists of aluminum (metal) and alumina (ceramic) with the following properties[19,27,28]

Ec=380 × 109N/m2, ρc=3800 kg/m3, αc=7.4 × 10− 6 ◦C− 1, Kc=10.4 W/mK

Em=70 × 109N/m2, ρm=2702 kg/m3, αm=23 × 10−6 ◦C−1, Km=204 W/mK

v = 0.3

and the ratio of geometric parameters of the selected plate a/b = 1, a/h = 20.

Table 1presents a comparison of the fundamental frequency parameter established in this paper withthe result of Ungbhakorn and Wattanasakulpong [20] They used the energy function and choosingresults under the type of moving position function for study FromTable 1, it can be seen that the presentvalues are not significantly different from the result in [20]

Table 2showing the effects of the group numbers of (m, n) and the volume coefficient ratio N on natural frequencies We can see that when the volume coefficient ratio N is increased while the natural frequencies decrease, and the value of the (m, n) is increased while the natural frequencies increases.

Table 2also shows that the lowest natural frequency corresponds mode (m, n) = (1, 1).

The effects of uniform increasing temperature to natural frequencies are shown inTable 3 and

Figure 2 We can see that the increasing of temperature makes the decreasing of frequencies and withthe same temperature variation, the uniform increasing temperature has the smaller natural frequenciesthan the nonlinear temperature transferring through the thickness of the plate.Table 3also shows theeffect of the elastic foundations: when κ1, κ2increase, they lead to the increase of natural frequencies

Figures 3and 4illustrate the effect of geometric factors of the FGM plate on nonlinear dynamic

response in the case of all FM edges with N = 1, (m, n) = (1, 1), ε = 0.1, 1T = 0 and without elastic

Table 1. Comparison of fundamental frequency parameter γ = ωhqρc

E c for Al/Al 2 O 3 square plates without elastic foundations and [a/b = 1, (m, n) = (1, 1), and 1T = 0].

Figures 3and 4illustrate the effect of geometric factors of the FGM plate on nonlinear. .. 8a

and specific expression of coefficients H are given in Appendix.

Numerical results and discussion

Consider a FGM plates acted on by an uniformly distributed... nonlinear dynamic

response in the case of all FM edges with N = 1, (m, n) = (1, 1), ε = 0.1, 1T = and without elastic< /i>

Table 1. Comparison of fundamental