Sound radiation of functionally graded materials plates in thermal environment

Composite Structures 144 (2016) 165–176

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Composite Structures journal homepage: www.elsevier.com/locate/compstruct

Sound radiation of functionally graded materials plates in thermal environment Tieliang Yang a, Weiguang Zheng b, Qibai Huang a,⇑, Shande Li a a b

State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China Electromechanical Engineering College, Guilin University of Electronic Technology, Guilin 541000, People’s Republic of China

a r t i c l e

i n f o

Article history: Available online 27 February 2016 Keywords: Vibro-acoustic response Functionally graded materials Temperature dependent material properties Thermal effects First-order shear deformation plate theory

a b s t r a c t In this paper, the vibration and sound radiation characteristics of functionally graded materials (FGM) plates subjected to thermal environment are investigated analytically. The FGM plate is made of a mixture of metal and ceramic, and the material properties are assumed to be temperature dependent and vary continuously through the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of the FGM plate subjected to thermal environment are derived based on the first-order shear deformation plate theory through Hamilton’s principle. The sound radiation of the FGM plate is calculated with Rayleigh integral. Comparisons of the present results with those of solutions available in literature are made and good agreements are achieved. Finally, the effects of temperature dependent material properties, material distribution, thermal field and temperature rise on the sound radiation of the FGM plates are studied. It is found that a dramatic discrepancy will occur if temperature dependent material properties are not taken into account. The material distribution, temperature rise and temperature fields also have significant effects on the vibro-acoustic response of the FGM plates. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded materials (FGM) are microscopically inhomogeneous materials in which material properties vary continuously and smoothly in the preferred directions, generally in the thickness direction [1–3]. This is usually achieved by a gradual variation of the volume fraction of the constituent phases. The concept of FGMs was originally proposed in 1984 by a group of material scientists in Japan [4] as ultrahigh temperature resistant materials for aerospace structural applications. The superior to conventional composite laminates, such as smaller thermal stresses, eliminating the stress concentrations, attenuation of stress waves, leads to a wider applications of FGMs in areas such as aircraft, space vehicles, nuclear plants and automotive applications [1]. In practical applications, FGM structures subjected to dynamic load internal or external will generate noise and radiate sound into surrounding medium, which may result in less comfort. Thus, knowledge of the acoustic behavior of a FGM plate is essential for the design of FGM structures. On the other hand, the radiated sound carries useful information of the FGM structures that can

⇑ Corresponding author. Tel.: +86 27 87557664. E-mail address: [email protected] (Q. Huang). http://dx.doi.org/10.1016/j.compstruct.2016.02.065 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.

be used for non-destructive evaluation or estimation of the material properties. Therefore, the investigation of sound radiation of FGM structures is of great importance from the academic or engineering point of view. Plates are one of the most widely used structural components in industrial applications. Sound radiation from plate structures is a practical engineering problem that has been studied extensively. However, in comparison with the extensive research works on the sound response characteristics of the laminated composite plates or multilayered plates (see for example Refs. [5–7]), only limited works can be found for FGM plates. Chandra et al. [8] analytically studied the vibro-acoustic response and sound transmission loss characteristics of FGM plates based on a simple first-order shear deformation theory. Yang et al. [9] presented a formulation of sound radiation of FGM plates based on three dimensional elasticity of theory and the state space method. It should be noted that the thermal effects are not considered in the aforementioned papers [8,9]. The FGM structures are always used in extreme thermal environment, and the temperature changes caused by the thermal environment may change the material properties, structure configuration and the stress state etc. [10]. Those changes affect the dynamic characteristics (also the sound radiation) of the FGM plate, which may become different

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from those of the FGM plate in non-thermal environment. Therefore, it is necessary to consider the thermal effects when dealing with the sound radiation of FGM plates in thermal environment. The early literature focused on the structure dynamic characteristics under thermal environment can date back to 1950s [11,12]. In recent years, the structural dynamic and acoustic radiation characteristics considering the thermal effect bring about some new problems with the increasing importance of the thermal environment of space vehicles [10]. Jeyaraj et al. [13] investigated the thermal effects on the vibration and acoustic response characteristics of a fiber-reinforced composite plate in a thermal environment with the use of the finite element method (FEM) and boundary element method (BEM). Geng et al. [10,14,15] carried out several investigations on the dynamic and acoustic responses of a simply supported or clamped rectangular plate in thermal environments. Zhao et al. [7] studied the vibration and acoustic response characteristics of orthotropic laminated composite plate in a hygroscopic environment. Li and Yu [16] studied the vibration and acoustic responses of the sandwich panels in a high temperature environment based on the piecewise low order shear deformation theory. In the aforementioned literature, the environment temperature change is not great, and the temperature dependent material properties were not considered. Kumar et al. [17] carried out parametric studies on the prediction of vibro-acoustic response from an elliptic disc made up of functionally graded material. With the use of commercial software, the vibration response and acoustic response of the FGM elliptic disc were obtained in FEM and BEM respectively. On the other hand, the literature on vibration and buckling response of FGM plates in thermal environment are extensive. Huang and Shen [18] investigated the nonlinear vibration and dynamic response of functionally graded material plates in thermal environments based on the higher-order shear deformation plate theory. Yang and Shen [19] studied the free and forced vibration of initially stressed functionally graded plates in thermal environment based on Reddy’s higher order shear deformation plate theory. Kim [20] developed an theoretical method to study the vibration characteristics of initially stressed FGM rectangular plate in thermal environments, and the temperature dependence of the material properties were considered. Shariat and Eslami [21] dealt with the mechanical and thermal buckling of rectangular thick functionally graded plates based on third order shear deformation plate theory, and it was found that the classical plate theory over-predicts the critical buckling loads, especially for thick plates. Malekzadeh et al. [22] studied the free vibration of FGM annular plates subjected to thermal environment based on the 3D elasticity theory and differential quadrature method, and it is found that the temperature-dependence of the material properties have significant effects on the natural frequency parameters. Malekzadeh and Monajjemzadeh [23] investigated the dynamic response of FGM plates in thermal environment under a moving load based on based on the first-order shear deformation theory. More recently, Pandey and Pradyumna [24] presented a layerwise finite element formulation for the dynamic analysis of FGM sandwich plates with nonlinear temperature variation along the thickness and the FGM having temperature dependent material properties. From the above literature survey, it is found that the works available in the literature with respect to vibro-acoustic response of FGM plates in thermal environment are limited. Motivated by this fact, in this paper, the vibration and acoustic response of FGM plates in thermal environment are investigated. The governing equations of the FGM plate subjected to thermal environment are derived based on the first-order shear deformation plate theory through Hamilton’s principle, and the acoustic response of the FGM plate is obtained with the use of the Rayleigh integral.

Accuracy of the results is examined by comparing the obtained results of the present formulation with that available in the literature. Finally, some parametric studies are conducted to investigate the acoustic characteristics of FGM plates in thermal environment. 2. Theoretical formulation 2.1. Geometry and material properties definition Consider a rectangular FGM plate of length a, width b and uniform thickness h in a Cartesian coordinate system ðx; y; zÞ, as shown in Fig. 1. The FGM plate is made of a mixture of two material phases, for example, a metal and a ceramic, and the material properties are assumed to vary continuously through the thickness direction according to the power law distribution. Based on the mixture rule, the effective material properties P(z) such as Young’s modulus E, density q, Poisson’s ratio m, thermal conductivity k, and thermal expansion a are expressed in terms of the material properties and volume fractions of constituents [25]

PðzÞ ¼ Pc V c þ P m V m

ð1Þ

where Pm and Pc denote the specific material properties of the metallic and ceramic constituents, respectively, and Vm and Vc represent the volume fractions of the metallic and ceramic constituents, respectively. By applying the power law distribution, the volume fractions of ceramic and metal are assumed as [25]

Vc ¼


z h

þ 0:5

N

;

V c þ V m ¼ 1 ð0:5h 6 z 6 0:5hÞ

ð2Þ

where N denotes the power-law index that takes a non-negative real number. Eq. (2) indicates that the top surface is the ceramicrich surface, whereas the metal-rich surface is at the opposite side. Substituting Eq. (2) into Eq. (1) yields


z N PðzÞ ¼ Pm þ ðPc  Pm Þ þ 0:5 h

ð3Þ

Then the effective Young’s modulus E, density q, Poisson’s ratio m, thermal conductivity k and thermal expansion a can be expressed as

N þ 0:5 þ Em h
z N aðzÞ ¼ ðac  am Þ þ 0:5 þ am h
z N qðzÞ ¼ ðqc  qm Þ þ 0:5 þ qm h
z N aðzÞ ¼ ðac  am Þ þ 0:5 þ am h
z N kðzÞ ¼ ðkc  km Þ þ 0:5 þ km h EðzÞ ¼ ðEc  Em Þ


z

ð4Þ

FGM are widely used in high temperature applications, and the material properties change at high temperature and are nonlinear

z

y b

h x

o a Fig. 1. Geometry and coordinates of the FGM plate.

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T. Yang et al. / Composite Structures 144 (2016) 165–176

function of temperature. To predict the behavior of FGMs under high temperature more accurately, it is necessary to consider the temperature dependent material properties. In this paper, it is assumed that the Young’s moduli E, Poisson’s ratio m, thermal expansion coefficient a of the FGM plate are temperature dependent, whereas mass density q and thermal conductivity k are independent of the temperature. For the temperature dependent material properties, the corresponding properties are given by [26]

PðTÞ ¼ P0 ðP1 T 1 þ 1 þ P1 T þ P 2 T 2 þ P 3 T 3 Þ

ð5Þ

where T is the temperature at an arbitrary material point of the plate T ¼ T 0 þ DT; T0 is the initial uniform temperature T0 = 300 K (where the plate is assumed to be stress free), and DT denotes the temperature change, P0, P1, P1, P2, and P3 are the temperature coefficients which are unique to the constituent materials. The temperature coefficients for some ceramics and metals are given in Table 1.

  d dT kðzÞ ¼0  dz dz

Consider the thermal boundary conditions obtained usually in the literature

T ¼ T t at z ¼ h=2 and T ¼ T b at z ¼ h=2

 
z Nþ1 ðT t  T b Þ
z ktb þ 0:5  þ 0:5 C h ðN þ 1Þkb h
z 2Nþ1
z 3Nþ1 k2tb k3tb þ þ 0:5 þ 0:5  ð2N þ 1Þk2b h ð3N þ 1Þk3b h # 4

z 5Nþ1 4Nþ1 ktb z k5tb þ 0:5 þ 0:5 þ  ð5N þ 1Þk5b h ð4N þ 1Þk4b h

TðzÞ ¼ T b þ

ð10Þ

It is assumed that no heat generation source exists within the plate, and the temperature variation occurs in the thickness direction only and one-dimensional temperature field is considered to be constant in the xy-plane [18,20,27]. Three cases of temperature change across the thickness of the plate are considered, i.e., uniform temperature rise, linear temperature rise and nonlinear temperature rise [27]. For uniform temperature rise, the temperature field is expressed as

T ¼ T 0 þ DT

ð6Þ

where T 0 is the initial uniform temperature T 0 ¼ 300 K, and DT denotes the temperature change. For linear temperature rise, the temperature field is expressed as


z

h

ð9Þ

Then the temperature change is defined as DT ¼ T t  T b , and the solution of Eq. (8) can be obtained by using the polynomial series solution [27,28]

2.2. Temperature field

TðzÞ ¼ T b þ DT

ð8Þ

 þ 0:5

ð7Þ

where Tb is the temperature at the bottom surface of the plate, and DT ¼ T t  T b with Tt the temperature at the top surface of the plate. For nonlinear temperature rise, the temperature distribution along the thickness can be obtained by solving the following steady-state heat transfer equation through the thickness of the plate

with kt ¼ kc ; kb ¼ km ; ktb ¼ kt  kb , and

C ¼1 

ktb k2tb k3tb k4tb þ  þ 2 3 ðN þ 1Þkb ð2N þ 1Þkb ð3N þ 1Þkb ð4N þ 1Þk4b

k5tb ð5N þ 1Þk5b

2.3. Governing equations In this study, the governing equations of the FGM plate in thermal environment is derived by utilizing the first-order shear deformation plate theory according to which the displacement field at any point of the plate can be written as [25]

uðx; y; z; tÞ ¼ u0 ðx; y; tÞ þ z/x ðx; y; tÞ

v ðx; y; z; tÞ ¼ v 0 ðx; y; tÞ þ z/y ðx; y; tÞ

ð11Þ

wðx; y; z; tÞ ¼ w0 ðx; y; tÞ where u; v and w are the displacement components along the x; y and z coordinates, respectively, u0 ; v 0 and w0 are the displacement components of the middle plane along the x; y and z coordinates, respectively and /x and /y are the rotations alone x and y coordinates.

Table 1 Temperature dependent material coefficients for ceramics and metals [18,20,22]. Material

Properties

P0

P1

P1

P2

P3

Ti-6Al-4V

E (Pa)

122.7  109 0.2888 4420 7.43  106 6.10

0 0 0 0 0

4.605  104 1.108  104 0 7.483  104 0

0 0 0 3.621  107 0

0 0 0 0 0

132.2  109 0.333 3657 13.3  106 1.78

0 0 0 0 0

3.805  104 0 0 1.421  103 0

6.127  108 0 0 9.549  107 0

0 0 0 0 0

201.04  109 0.28 8166 12.33  106 12.04

0 0 0 0 0

3.079  104 0 0 8.086  104 0

6.534  107 0 0 0 0

0 0 0 0 0

348.43  109 0.28 2370 5.872  106 9.19

0 0 0 0 0

3.07  104 0 0 9.095  104 0

2.160  107 0 0 0 0

8.946  1011 0 0 0 0

t q (kg/m3) a (1/K) k (W/mK) ZrO2

E (Pa)

t q (kg/m3) a (1/K) k (W/mK) SUS304

E (Pa)

t q (kg/m3) a (1/K) k (W/mK) Si3N4

E (Pa)

t q (kg/m3) a (1/K) k (W/mK)

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T. Yang et al. / Composite Structures 144 (2016) 165–176

The nonzero linear strains associated with the displacements are

exx ¼

  @/y @u0 @/ @v 0 @ v 0 @u0 @/x @/y þ z x ; eyy ¼ þz ; cxy ¼ þ þz þ @x @x @y @y @x @y @y @x

cyz ¼ /y þ

@w0 @w0 ; c ¼ /x þ @y xz @x

Substituting Eq. (11) into Eq. (18) and the variation of virtual kinetic energy can be given as

Z

a

dT ¼ 0


1 0  @/ @d/ 0 @du0 0 @dw0 x @d/x I0 @u þ @@tv 0 @d@tv 0 þ @w þ @ty @t y þ I2 @/ @t @t @t @t @t @t B C @ AdA
 @/y @dv 0 @/x @du0 @ v 0 @d/y 0 @d/x þI1 @u þ þ þ @t @t @t @t @t @t @t @t

ð19Þ

ð12Þ

with (I0, I1, I2) are the mass moment of inertias which are defined as

where exx ; eyy ; cxy ; cyz and cxz are the strain components. The linear constitutive relations are

8 > > > > > > > <

9 38 2 rx 9 0 0 > ex > c11 c12 0 > > > > > > > 7 6 > > > ey > ry > 0 0 7> > > > 6 c12 c22 0 > > > = 6 = 7< 7 c 6 ryz ¼ 6 0 0 c44 0 0 7 yz > > > 7> 6 > > > > >c > > > > > 6 > 0 0 c55 0 7 > > 5> xz > > rxz > > > 4 0 > > > > > : : ; ; cxy rxy 0 0 0 0 c66

Z ðI0 ; I1 ; I2 Þ ¼

ð13Þ

0:5h

qðzÞð1; z; z2 Þdz

ð20Þ

The variation of work done by the external forces can be expressed as

Z

dV 1 ¼ 

ð21Þ

qdwdA A

where cij are the elastic coefficient which is a function of the z-coordinate and temperature and are given by

Eðz; TÞ v ðz; TÞEðz; TÞ ; c12 ¼ c21 ¼ ; 1  v ðz; TÞ2 1  v ðz; TÞ2 Eðz; TÞ ¼ c66 ¼ 2ð1 þ v ðz; TÞÞ

The variation of work done by the temperature change can be expressed as

dV 2 ¼

c11 ¼ c22 ¼ c44 ¼ c55

0:5h

 Z  @w0 @dw0 @w0 @dw0 @w0 @dw0 dA NTxx þ N Txy þ NTyy @x @x @x @y @y @y A

ð22Þ in which

Z

0:5h



 Eðz; TÞ dz; 1  mðz; TÞ

According to the Hamilton’s principle, the dynamic equations of the FGM plate can be derived by

ðNTxx ; NTyy Þ ¼ 

Z

Substituting Eqs. (16), (19), (21) and (22) into Eq. (14) and collecting the coefficient of du0 ; dv 0 ; dw0 ; d/x and d/y , the following equations of motion can be obtained

t1

dðU þ V 1 þ V 2  TÞdt ¼ 0

ð14Þ

t0

where t0 and t1 denote two arbitrary time values, d is the variational operator, T is the kinetic energy of the system, U is the potential energy of the system, V1 is the potential energy done by the external load q and V2 is the potential energy induced by the thermal effects. The virtual strain energy can be calculated as

Z

dU ¼

rij deij dV V

Z ¼

V

ðrx dex þ ry dey þ rxy dcxy þ rxz dcxz þ ryz dcyz ÞdV

ð15Þ


1 @d/ 0 x 0 Nxx @du þ M xx @d/ þ Nyy @d@yv 0 þ Myy @y y þ Nxy @du þ @d@xv 0 @x @x @y B C dU ¼ @

 AdA  @d/y @d/x @dw0 @dw0 V þMxy @y þ @x þ Q x d/x þ @x þ Q y d/y þ @y 0

ð16Þ where Nij, Mij and Qij are the stress resultants defined by

Z

ðNxx ; Nyy ; Nxy Þ ¼

h 2

Z

Z

h 2

h2

h 2

2h

ðrxx ; ryy ; sxy Þzdz

ð17Þ

ðsyz ; sxz Þdz

where j2 is the shear correction factor. The kinetic energy of the plate can be calculated as

 2 @ui dV @t V  2  2  2 ! Z Z 0:5h 1 @u @v @w dzdA ¼ qðzÞ þ þ 2 A 0:5h @t @t @t

T¼

1 2

Z

qðzÞ

ð18Þ

ð23Þ

@Nxx @Nxy @ 2 u0 @ 2 /x þ ¼ I 0 2 þ I1 @x @y @t @t 2

dv 0 :

@ 2 /y @Nyy @Nxy @2v 0 þ I1 þ ¼ I0 2 @y @x @t @t 2

dw0 :

@Q x @Q y @ 2 w0 @ 2 w0 @ 2 w0 @ 2 w0 þ NTyy þ 2NTxy þ þ NTxx þ q ¼ I0 2 2 @x @y @x @y @x@y @t2

d/x :

@M xx @M xy @2/ @ 2 u0 þ  Q x ¼ I 2 2x þ I 1 2 @x @y @t @t

d/y :

@ 2 /y @M xy @M yy @2v 0 þ I1 þ  Q y ¼ I2 2 @x @y @t @t 2 ð24Þ

Substituting Eq. (17) into Eq. (24), Eq. (24) can be rewritten with respect to the displacement components

@ 2 /y @ 2 u0 @2v 0 @ 2 /x þ A12 þ B12 þ B11 2 2 @x @x@y @x @x@y ! ! 2 2 2 2 @ u0 @ v 0 @ /x @ /y @ 2 u0 @ 2 /x þ B ¼ I 0 2 þ I1 þ A66 þ þ 66 2 2 @y @x @y @y @x @y @t @t 2 ! @ 2 u0 @2v 0 @ 2 u0 @ 2 v 0 dv 0 : A12 þ A66 þ A22 þ 2 @x@y @y @x@y @x2 ! @ 2 /y @ 2 /y @ 2 hx @ 2 hy @ 2 /x @2v 0 ¼ I þ I þ 2 þ B12 þ B22 þ B66 0 1 @x @y @x @x@y @y2 @t2 @t 2 ! ! 2 @ 2 w0 @/x @ 2 w0 @/y T @ w0 2 þ þ N dw0 : j2 A44 þ j A þ 55 xx @x2 @x @y2 @y @x2 du0 : A11



rxx ; ryy ; sxy dz

2h

ðM xx ; M yy ; Mxy Þ ¼ ðQ y ; Q x Þ ¼ j2



NTxy ¼ 0

du0 :

Substituting Eqs. (11)–(13) into Eq. (15) yields

Z

0:5h

aðz; TÞDT

þ NTyy

@ 2 w0 @ 2 w0 @2w þ 2NTxy þ q ¼ I0 2 2 @y @x@y @t

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T. Yang et al. / Composite Structures 144 (2016) 165–176

@ 2 /y @ 2 u0 @2v 0 @ 2 /x þ B þ D þ D 12 11 12 @x2 @x @y @x2 @x@y ! ! 2 2 2 2 @ /y @ u0 @ v 0 @ /x þ B66 þ þ þ D66 @y2 @x @y @y2 @x @y   2 2 @w0 @ u0 @ /x þ /x ¼ I1 2 þ I2  j2 A44 @x @t @t 2 ! 2 2 2 @ u0 @ v0 @ u0 @ 2 v 0 d/y : B12 þ B þ B22 þ 66 @x@y @y2 @x @y @x2 ! @ 2 /y @ 2 hx @ 2 hy @ 2 /x þ D66 þ 2 þ D12 þ D22 @x @y @x @x @y @y2   @ 2 /y @w0 @2v 0  j2 A55 þ I2 þ /y ¼ I1 2 @y @t @t2

s45 ¼ ðD12 þ D66 Þðnp=bÞðmp=aÞ;

d/x : B11

s52 ¼ B66 ðmp=aÞ þ B22 ðnp=bÞ ;

s51 ¼ ðB12 þ B66 Þðnp=bÞðmp=aÞ; s53 ¼ j2 A55 ðnp=bÞ;

s54 ¼ ðD12 þ D66 Þðnp=bÞðmp=aÞ; 2

s55 ¼ D66 ðmp=aÞ2 þ D22 ðnp=bÞ þ j2 A55

2.4. Transverse load In this paper, only the transverse load on the surface of the plate is considered and the load q(x,y) is assumed to be expanded into a double Fourier series form [30]

ð25Þ

qðx; yÞ ¼

1 X 1 X

qmn sinðmpx=aÞ sinðnpy=bÞ

ð30Þ

m¼1 n¼1

where qmn is the load coefficient which is given as

where

Z ðAij ; Bij ; Dij Þ ¼

0:5h

ð1; z; z Þcij dz

ð26Þ

2

0:5h

In this paper, only the simply supported boundary conditions are considered, and the boundary conditions are [29]

Nxx ¼ v ¼ w ¼ Mxx ¼ 0;

at x ¼ 0; a

u ¼ Nyy ¼ w ¼ M yy ¼ 0;

at y ¼ 0; b

ð27Þ

The state variables satisfying the simply supported boundary conditions of Eq. (27) are assumed as the following form

8 9 8  9 u > u cosðmpx=aÞ sinðnpy=bÞ > > > > > > > > > > > > > > > > > >  sinðmpx=aÞ cosðnpy=bÞ > v v > > > > > > > > 1 X 1 < < = X =  sinðmpx=aÞ sinðnpy=bÞ expðixtÞ w ¼ w > > > > > > m¼1 n¼1 > > > >  x cosðmpx=aÞ sinðnpy=bÞ > > > > > /x > / > > > > > > > > > > > > : ; : ;  y sinðmpx=aÞ sinðnpy=bÞ /y /

ð28Þ

pffiffiffiffiffiffiffi  x ; /Þ  are  ; v ; w;  / where i ¼ 1; x is the circular frequency, and ðu undetermined coefficients. By substituting Eq. (28) into Eq. (25), the following equation can be derived

ðS  x2 MÞU ¼ Q  where U ¼ ½ u

2

2

2

s11

6 6 s21 6 S¼6 6 s31 6 4 s41 s51

s12

v

ð29Þ  w

s13

x /

s14

s22 s32

s23 s33

s24 s34

s42

s43

s44

s52

s53

s54

 y  , and Q ¼ ½ 0 / T

s15

3

7 s25 7 7 s35 7 7; 7 s45 5 s55

2

I0

6 60 6 M¼6 60 6 4 I1 0

0

0

q

0

0

0 T

I1

0

I0 0

0 I0

0 0

0

0

I2

I1

0

0

3

7 I1 7 7 07 7 7 05 I2

2

s12 ¼ ðA12 þ A66 Þðnp=bÞðmp=aÞ;

s14 ¼ B11 ðmp=aÞ2 þ B66 ðnp=bÞ ;

2

s15 ¼ ðB12 þ B66 Þðnp=bÞðmp=aÞ;

s21 ¼ ðA12 þ A66 Þðnp=bÞðmp=aÞ;

s22 ¼ A66 ðmp=aÞ2 þ A22 ðnp=bÞ ;

s13 ¼ 0; 2

s23 ¼ 0; s24 ¼ ðB12 þ B66 Þðnp=bÞðmp=aÞ; s31 ¼ 0;

2

s25 ¼ B66 ðmp=aÞ2 þ B22 ðnp=bÞ ;

s32 ¼ 0; 2

2

s33 ¼ j ðA44 ðmp=aÞ2 þ A55 ðnp=bÞ Þ þ NTxx ðmp=aÞ2 þ NTyy ðnp=bÞ ; 2

s34 ¼ j2 A44 ðmp=aÞ; s35 ¼ j2 A55 ðnp=bÞ;

Z

a

Z

b

qðx; yÞ sin 0

0


mpx
npx sin dxdy a b

ð31Þ

2.5. Sound radiation power The vibrating plates are assumed to be placed in an infinite rigid baffle when dealing with the sound radiation of the plates, which, of course, differs from the actual situation in practice, and some authors concentrated on sound radiation from unbaffled plates [31,32]. One of the authors of this paper studied the sound radiation characteristic from un-baffled rectangular plates [33]. More recently, Putra and Thompson [34,35] also investigated this problem. However, the assumption of an infinite rigid baffle simplifies the calculations of the sound radiation. This is mainly due to the fact that the velocity field equals to zero everywhere except for the plate surface with the assumption of an infinite rigid baffle, which allows the use of the Rayleigh integral instead of the full Kirchhoff–Helmholtz integral equation [36]. Therefore, the assumption of an infinite rigid baffle is also obtained here for the convenience in studying the main trends of sound radiation of FGM plates in thermal environment. For a plate set in an infinite rigid baffle, the acoustic pressure at any field point r can be expressed in terms of surface complex _ s Þ according to Rayleigh integral [36,37] velocity wðr

jxq0 2p

Z

S

_ sÞ wðr

0

expðjkr Þ dS r0

ð32Þ

Where pðrÞ is the complex pressure amplitude at location r, r0 ¼ jr  rs j is the distance between a surface point rs and a field point r, k is the acoustic wavenumber and k = x/c0 with c0 being the speed of sound in the medium, q0 is the density of the acoustic medium. When subjected to external excitation, the FGM plate will be stimulated and radiate noise into the surrounding acoustic medium. It is convenient to characterize the sound radiated from a vibrating structure by a single global quantity such as sound radiation power. There are two different approaches generally used to determine the sound power. The first is to integrate the acoustic intensity over the vibrating surface, and the second is to integrate the squared pressure on a hemisphere in the far field [38]. Only the first one is adopted in this work. The radiated sound power can be obtained by integrating the acoustic intensity over the surface of the plate [36]

 ¼ W

2

s41 ¼ B11 ðmp=aÞ2 þ B66 ðnp=bÞ ;

Z

_  ðrs ÞpðrÞÞdS 0:5Reðw

ð33Þ

⁄

denote the real part and the complex

S

s42 ¼ ðB12 þ B66 Þðnp=bÞðmp=aÞ; s43 ¼ j2 ðA44 ðmp=aÞ;

4 ab

pðrÞ ¼

and sij are given as

s11 ¼ A11 ðmp=aÞ2 þ A66 ðnp=bÞ ;

qmn ¼

2

s44 ¼ D11 ðmp=aÞ2 þ D66 ðnp=bÞ þ j2 A44 ;

where Re and superscript conjugate, respectively,

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T. Yang et al. / Composite Structures 144 (2016) 165–176

In this paper, a primitive numerical scheme [8,39] is used to solve the Rayleigh integral and obtain the sound radiation power. The plate is divided into M  N equal size rectangular elements, and the normal velocity is assumed to be constant across the rectangular elements. This approximation is sufficiently accurate if the pffiffiffiffiffiffi characteristic length of the elemental radiator DS is much smaller than half of the minimum acoustic wavelength, with DS the area of the elemental radiator. Then the radiated sound power in Eq. (33) can be formulated as [8]

 ¼w _ Hn ðrÞRw _ n ðrÞ W

ð34Þ

_ n ðrÞ is the vector with normal surface velocities of the elewhere w mental radiators, R is the radiation resistance matrix. The radiation resistance matrix of the plate radiators placed in an infinite baffle can be written as

2

1

6 6 sinðkr21 Þ 6 kr21 q 0 x DS 6 6 R¼ 4pc0 6 6 .. 6 . 4 2

2

sinðkrn1 Þ kr n1

sinðkr12 Þ kr 12



sinðkr1n Þ 3 kr1n

1



sinðkr2n Þ 7 7 kr2n 7

.. .

..

sinðkr n2 Þ kr n2

7

.

.. .



1

7 7 7 7 5

ð35Þ

The radiated sound power is usually written in the form of sound power level in decibel, which is defined by

 W  0Þ SPL ¼ 10 logðW=

ð36Þ

 0 is the reference power and W  0 ¼ 1  1012 W: with W 2.6. Sound radiation efficiency The sound radiation efficiency is a commonly used parameter for sound radiation characteristics of plate structures and used as a measure for how well the vibrating object radiates sound. The sound radiation efficiency is generally defined as [36,37]

r¼

 W

q0 c0 Shw_ 2 i

ð37Þ

which q0 is the fluid density, c0 is the sound speed in that fluid, S is  is the sound radiation power of the plate, the plate surface area, W _ 2 i is the spatially averaged mean-square normal velocity and hw across the total surface of the plate which is defined as

_ 2i ¼ hw

1 2S

Z

_ 2 dS jwj

ð38Þ

S

Following the elemental radiators approach, the radiation efficiency in Eq. (37) can be written as [8]

r¼

_ n ðrÞ _ Hn ðrÞRw w q0 c0 Sw_ Hn ðrÞNw_ n ðrÞ

ð39Þ

_ n ðrÞ is the vector with normal surface velocities of the elewhere w mental radiators which is the same as that in Eq. (34), and N = (1/2N)I with I the identity matrix and N the number of the elemental radiators. 3. Validation study 3.1. Validation of the structural model In order to evaluate the accuracy of the present formulation for the natural frequency of FGM plates in the thermal environment, the results obtained by the present formulation are compared with those available in the literature. A simply supported isotropic

square FGM plate is considered for the study, which is taken from the work of Huang and Shen [18] and also used as a validation case in Pandey and Pradyumna [24]. The FGM plate is made of silicon nitride (Si3N4) and stainless steel (SUS304), whose temperature dependent material properties are given in Table 1. The geometric properties of the FGM plate are given as a = b = 0.2 m and h = 0.025 m. The thermal environment is considered to be nonlinear temperature variation along the thickness. For simplicity, the non-dimensional natural frequency parameter is considered, which is defined as

- ¼ xða2 =hÞðð1  t2m Þqm =Em Þ

1=2

ð40Þ

where qm, Em and tm are chosen to be the values of stainless steel at the reference temperature T0 = 300 K. Table 2 compares the results of the present work with that of Huang and Shen [18] as well as Pandey and Pradyumna [24]. It should be noted that the present work is based on the first-order shear deformation theory, the work of Huang and Shen [18] is based on the third-order shear deformation theory, and the work of Pandey and Pradyumna [24] is based on the layerwise finite element formulation. As shown in Table 2, the results obtained by the present work are in good agreement with that of Huang and Shen [18] as well as Pandey and Pradyumna [24]. 3.2. Validation of the sound radiation power calculation After carrying out the validation of the structural model for FGM plate in thermal environment, we next validate the acoustic model of the present formulation. Since there are no available results for sound radiation from FGM plates subjected to thermal environment, to validate the present model, two case studies, sound radiation from isotropic Aluminum plate and FGM plate are taken up. The first example is sound radiation from a rectangular Aluminum plate, which is taken from Geng et al. [10]. The size of the plate is 0.4 m  0.3 m  0.01 m. It should be noted that the material properties are considered as temperature-independent, and the Young’s modulus, Poisson’s ratio, mass density and thermal expansion coefficient are assumed to be 70 GPa, 0.3, 2700 kg/m3 and 2.3e5/K, respectively [10]. In addition, a loss factor of 0.001 is considered and the temperature rise is uniform. Table 3 shows the comparison of the first 5 natural frequencies of the plate subjected to temperature rise of DT ¼ 0  C; DT ¼ 15  C and DT ¼ 45  C : The comparison of the results of the sound radiation power of the plate obtained by the present formulation and Geng et al. [10] is shown in Fig. 2. The results of sound power in watts in Geng et al. [10] has been converted to dB (Ref. 1012 W) to maintain uniformity of units. As can be seen from Table 3 and Fig. 2, the results obtained from the present formulation and the model of Geng et al. [10] are in good agreement. The discrepancy between the two predictions are mainly due to the fact that the classic plate theory was obtained in Geng et al. [10], whereas the present formulation is based on the first-order shear deformation theory. As is well known that the shear deformation effect is not considered in the classic plate theory, which leads to overestimating the natural frequencies of the plate. In addition, It should be noted that the analysis step is 20 Hz in Geng et al. [10], which results into losing some amplitude of the picks of the sound power level, and the analysis step is 1 Hz in the present formulation. The second example is sound radiation from FGM (Al/Al2O3) plates which is taken from Yang et al. [9]. It should be noted that the thermal effects are not considered and the material properties of the FGM plate are the same as that in Yang et al. [9]. The plate is assumed to be simply supported on all edges and with the dimensions of 0.5 m  0.4 m  0.003 m. Fig. 3 shows the comparison of

171

T. Yang et al. / Composite Structures 144 (2016) 165–176 Table 2 Comparisons the non-dimensional frequency parameter Mode

- ¼ xða2 =hÞðð1  t2m Þqm =Em Þ1=2 of the simply supported FGM plate subjected to different thermal conditions. Huang and Shen [18]

Pandey and Pradyumna [24]

Present

(1, 1)

(1, 2)

(1, 1)

(1, 2)

(1, 1)

(1, 2)

N 0 0.5 1. 2.0 1

12.495 8.675 7.555 6.777 5.405

29.131 20.262 17.649 15.809 12.602

12.410 8.577 7.537 6.757 5.450

29.358 20.273 17.809 15.698 12.792

12.505 8.607 7.545 6.774 5.410

29.248 20.122 17.634 15.827 12.652

Tt = 400 K, Tb = 300 K

0 0.5 1. 2.0 1

12.397 8.615 7.474 6.693 5.311

29.083 20.215 17.607 15.762 12.539

12.190 8.514 7.411 6.665 5.312

28.960 20.042 17.623 15.89 12.812

12.170 8.372 7.339 6.595 5.274

28.815 19.827 17.378 15.604 12.489

Tt = 600 K, Tb = 300 K

0 0.5 1. 2.0 1

11.984 8.269 7.171 6.398 4.971

28.504 19.783 17.213 15.384 12.089

11.762 8.143 7.15 6.432 4.996

28.224 19.602 17.260 15.540 12.333

11.437 7.853 6.894 6.192 4.903

27.916 19.201 16.830 15.114 12.037

Tt = 300 K, Tb = 300 K

Table 3 The natural frequencies of an isotropic plate in thermal environment: comparisons between the present model with the model of Geng et al. [10].

3.3. Validation of the sound radiation efficiency calculation

Mode (m, n)

Method

DT = 0 °C

DT = 15 °C

DT = 45 °C

1 (1, 1)

Present Geng et al.

418.8 420.2

346.5 348.0

96.5 100.9

2 (2, 1)

Present Geng et al.

868.3 874.0

799.4 805.3

639.5 646.4

3 (1, 2)

Present Geng et al.

1215.8 1227.0

1147.6 1159.1

997.2 1009.8

In order to validate the formulation and the developed code for the calculation of sound radiation efficiency, a case for the sound radiation efficiency of a damped simply-supported aluminum plate (E = 71 GPa, q = 2700 kg/m3, m = 0.35 and damping loss factors g = 0.1) in air is obtained, which is taken from Arenas [39]. The plate size is 0.5 m  0.6 m  0.003 m, and subjected a unit amplitude concentrated force applied at point x = y = 0.02 m. As shown in Fig. 4, the calculated result is compared with that of Arenas [39] and a good agreement is achieved.

4 (3, 1)

Present Geng et al.

1610.8 1630.4

1542.9 1563.0

1397.4 1418.6

4. Results and discussions

5 (2, 2)

Present Geng et al.

1660.0 1680.8

1592.2 1613.4

1447.0 1469.5

Natural frequencies (Hz)

120

120

100

100 Sound power level [dB]

Sound power level [dB]

the results obtained by the present formulation and that based on the 3-D elasticity [9]. As the plate is very thin (a/h = 166.7), the difference between the results based on the first-order shear deformation theory and that based on 3-D elasticity is small. The close agreements between the present results and that based on 3-D elasticity [9] demonstrates the accuracy of the present formulation for sound radiation of FGM plates.

In this subsection, the developed formulation is thus deployed to carry out several parametric studies to examine the vibration and acoustic response of FGM plates in thermal environments. A rectangular ZrO2/Ti-6Al-4V FGM plate, simply supported on all edges with dimensions of 0.4 m  0.3 m  0.02 m is considered for the following detailed investigations. The temperature dependent material properties of ZrO2/Ti-6Al-4V are given in Table 1. The plate is assumed to be vibrating in air. For the sake of convenience, the air density is taken to be q0 = 1.21 kg/m3, and the speed of sound in the air is taken as c0 = 343 m/s. In addition, a damping loss factor of 0.01 is taken for the following calculations.

80 60 40

Present Geng et al.

20

80 60 40 20

(b)

(a) 0 0

500 1000 1500 Frequency [Hz]

2000

0 0

500 1000 1500 Frequency [Hz]

2000

Fig. 2. Comparison of sound radiation power of isotropic plate between the present model and Geng et al. [10]: (a) DT ¼ 15  C, (b) DT ¼ 45  C.

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T. Yang et al. / Composite Structures 144 (2016) 165–176

(a)

(b)

Fig. 3. Comparison of sound radiation power of FGM plate between the present model and the model based on 3-D elasticity [9]: (a) power-law index N ¼ 1; (b) power-law index N ¼ 5.

1

120

10

100 0

Sound power level [dB]

Radiation efficiency

10

-1

10

-2

10

Present Arenas J .P.

80 60 40 3X3 5X5

20

10X10

-3

10

1

2

10

10

3

4

10 Frequency [Hz]

10

Fig. 4. Comparison of sound radiation efficiency of a damped simply-supported aluminum plate obtained by the present formulation and that of Arenas [39].

Table 4 The critical buckling temperature of the FGM plate (ZrO2/Ti-6Al-4V, dimension size 0.4 m  0.3 m  0.02 m) with different material distribution under different thermal fields. Thermal field

Uniform Linear Nonlinear

20X20

0

Critical buckling temperature DTc (K) N=0

N = 0.2

N=1

N=5

N=1

418.31 894.21 894.21

417.51 884.84 990.92

421.02 885.33 1095.95

429.46 908.94 1045.53

424.81 908.98 908.98

The critical buckling temperature (DTcr) is an important parameter to be considered while investigating the vibro-acoustic response of plate structures under thermal environment. The critical buckling temperature is influenced by such as the material properties, the aspect ratio of the plate, etc. When the temperature exceeds the critical buckling temperature, the plate buckling will occur and the plate will not be that plane and more nonlinear factors should be considered for the sake of accuracy [10,16]. The present formulation in this paper is a linear method. Once the temperature load exceeds the critical buckling temperature, the present formulation in this paper is not appropriate to obtain the responses of the FGM plates for the sake of accuracy. Therefore, the temperature of the thermal environment considered in this paper is lower than the critical buckling temperature. The critical buckling temperature of the FGM plate under different thermal

-20 0

30X30

500

1000 1500 2000 Frequency [Hz]

2500

3000

Fig. 5. Convergence of sound radiation power with different numbers of elemental radiators.

fields is presented in Table 4. It is worthwhile to point out that the N = 0 and N = 1 are representatives of FGM plates of monoceramic and mono-metal plates respectively. As depicted in Eq. (34), a primitive numerical scheme (elementary radiators approach) is used to solve the Rayleigh integral and to obtain the sound radiation power. The number of elementary radiators has a significant effect on the accuracy of this method [8,39]. Therefore, it is worthwhile to express the used number of elementary radiators in detail. A convergence study with respect to the number of elementary radiators on the calculation of the sound radiation power is carried out and the results are shown in Fig. 5. As is illustrated in Fig. 5, the number of elementary radiators plays an important role in the sound power calculation, especially in relatively high frequency range. In addition, it is found that the difference between the results with the number of elementary radiators 10  10 (=100), 20  20 (=400) and 30  30 (=900) is small. For the sake of efficiency and accuracy, the number of elementary radiators 20  20 (=400) is taken for the following calculations. 4.1. Investigating the influence of temperature dependent material properties In this paper, the temperature dependent material properties of FGM plates are considered. In order to demonstrate the importance

173

120

100

100

80 60 40 20 0 -20 -40 0

Sound power level [dB]

Temperature independent Temperature dependent

Sound power level [dB]

120

80 60 40 20 0 -20

(a) 500

1000 1500 2000 Frequency [Hz]

-40 0

2500

120

120

100

100

Sound power level [dB]

Sound power level [dB]

T. Yang et al. / Composite Structures 144 (2016) 165–176

80 60 40 20 0 -20 -40 0

500

1000 1500 2000 Frequency [Hz]

2500

500

1000 1500 2000 Frequency [Hz]

2500

1000 1500 2000 Frequency [Hz]

2500

80 60 40 20 0 -20

(c)

(b)

(d)

-40 0

500

Fig. 6. Comparisons of sound radiation from FGM plates subjected to thermal environment with and without considering temperature-dependent material properties. (a) Temperature field DT = 100 K, (b) temperature field DT = 200 K, (c) temperature field DT = 300 K, (d) temperature field DT = 400 K.

120 100

Sound power level [dB]

of considering the temperature dependent material properties in sound radiation of FGM plates in thermal environment, the comparison of sound radiation from FGM plates with and without the temperature dependent material properties are conducted. The comparison of the sound radiation power level is showed in Fig. 6 The thermal field is assumed to be uniform temperature rise through the thickness, and the temperature rise of DT = 100 K, DT = 200 K, DT = 300 K and DT = 400 K are considered. As shown in Fig. 6, the difference between the results of the sound radiation power with and without considering the temperature dependent material properties is small when the temperature rise is small, however, the discrepancy of that will increase dramatically when the temperature rise increases. The peaks of sound power level correspond to the natural frequencies. From Fig. 6, it can be seen that the peaks of sound power level shift to lower frequency range when considering the temperature dependent material properties, which indicates that the natural frequencies of the FGM plates will be overestimated when the temperature dependent material properties are not taken into account.

80 60 40

N=0 N=0.2 N=1 N=5 N=

20 0 -20

0

500

1000

1500

2000

Frequency [Hz] 4.2. Investigating the influence of material distribution Fig. 7 illustrates the effects of material distribution on the sound radiation of FGM plates in thermal environment. Sound radiation of different FGM plates with different material distribution with the power-law index of N = 0, N = 0.2, N = 1, N = 5 and N = 1 are considered. The N = 0 and N = 1 are representatives of FGM plates of mono-ceramic and mono-metal plates respectively, N = 0.2, N = 1 and N = 5 are representatives of FGM plates with ceramic rich, linear and metal rich, respectively. The thermal condition is uniform thermal field DT = 300 K. It can be seen that the sound radiation power of functionally graded plates are influenced signif-

Fig. 7. Sound radiation of FGM plates in thermal environment with different material distribution.

icantly by the volume fraction distributions of the constituents. The sound power level increases when the power-law index increases at very lower frequencies. The peaks of sound power level shift towards lower frequency range with the increase of the power-law index. This is due to the fact that the decrease of the stiffness of the FGM with the increase of the power-law index N, which resulting in the decrease of corresponding natural frequencies.

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T. Yang et al. / Composite Structures 144 (2016) 165–176

4.3. Investigating the influence of thermal field In order to research the effects of thermal field on sound radiation of FGM plates, the variation of sound radiation power of FGM plates subjected to different thermal fields, i.e., uniform thermal field, linear thermal field and nonlinear thermal field are shown in Fig. 8. In addition, the temperature rises are all taken as DT = 300 K. As shown in Fig. 8, the difference between the sound radiation results of the FGM plates subjected to linear thermal field and the nonlinear thermal field is small; however, there is a distinct discrepancy between the results of the FGM plates subjected to the uniform thermal field and the linear thermal field or the nonlinear thermal field. The thermal field can affect the thermal stress as well as the material properties. The average temperature through the thickness is not the same. The linear thermal field DT = 600 K is also selected. As shown in the figure, the difference between the uniform thermal field DT = 300 K and linear thermal field DT = 600 K, which have an average temperature through thickness, is small. Another FGM plate (Si3N4/SUS304) subjected to different thermal field is taken up, and the results of sound radiation of FGM plates subjected to different thermal fields are plotted in Fig. 9. As shown in the figure, a similar conclusion can be drawn that the difference between the results of sound radiation from

Sound power level [dB]

150

100

50 linear T=300 K nonlinear T=300 K uniform T=300 K linear T=600 K

0

-50 0

500

1000 1500 2000 Frequency [Hz]

2500

Fig. 8. Sound radiation power of a FGM plate (ZrO2/Ti-6Al-4V) subjected to different temperature fields.

FGM plates subjected to linear and nonlinear thermal field is small, and that between the uniform thermal field and linear or nonlinear thermal field is distinct. Although the FGM plates subjected to linear thermal field DT = 400 K share a same average temperature through the thickness with the FGM plate subjected to uniform thermal field DT = 200 K, the corresponding peaks of FGM plates subjected to linear thermal field DT = 200 K shift towards to lower frequency compared with that of FGM plates subjected to uniform thermal field DT = 200 K. It can be concluded from the aforementioned detailed comparisons that the thermal field through the thickness has a significant importance on the sound radiation of the FGM plates, and in some not sensitive cases, the nonlinear thermal field can be replaced by the linear thermal field when dealing with the sound radiation from FGM plates in thermal field. 4.4. Investigating the influence of temperature rise Fig. 10 illustrates the effect of the temperature rise on sound radiation of FGM plates. The thermal field across the thickness is uniform and the temperature rise of DT = 0 K, DT = 100 K, DT = 200 K, DT = 300 K and DT = 400 K are considered. It is observed that the corresponding peaks of the sound power level shift towards to lower frequency domain when the temperature rise increase. This is due to the decrease of the corresponding natural frequencies with the increase of temperature rise. In order to express this point clearly, the first 5 natural frequencies and the first 5 natural frequency ratios of the FGM plates subjected to different temperature rise are shown in Figs. 11 and 12, respectively. The natural frequency ratio is taken as the ratio of the natural frequency under any temperature to that under the temperature of T0 = 300 K. As shown in Fig. 11, the natural frequencies decrease with the increase of the temperature rise. The temperature rise not only decreases the elastic modulus but also induces the internal compressive stresses, which will decrease of the rigidity of the plate and the corresponding natural frequencies decrease. In addition, it is also observed from Fig. 12 that the first natural frequency is much more sensitive to the temperature change than the other natural frequencies, as it will firstly approach zero with the increase of the temperature rise. 4.5. Investigating the thermal effects on the sound radiation efficiency and critical frequency In order to investigate the thermal effect on the sound radiation efficiency of FGM plate, the sound radiation efficiency of a FGM

120

140 120

80 60 40 20

linear T=200 K nonlinear T=200 K uniform T=200 K linear T=400 K

0 -20 -40 0

Sound power level [dB]

Sound power level [dB]

100

100 80 60 40

T=0 K T=100 K T=200 K T=300 K T=400 K

20 0 -20

500

1000 1500 2000 Frequency [Hz]

2500

Fig. 9. Sound radiation power of a FGM plate (Si3N4/SUS304) subjected to different temperature fields.

0

500

1000 1500 2000 Frequency [Hz]

2500

3000

Fig. 10. Sound radiation power of a FGM plate subjected to different uniform temperature rise.

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T. Yang et al. / Composite Structures 144 (2016) 165–176 1

6000

10

mode (1,1) mode (2,1) mode (1,2) mode (3,1) mode (2,2)

4000

0

Radiation efficiency

Natural frequency

5000

3000 2000 1000

10

-1

10

T=0 K T=50 K T=100 K

-2

10

0 2

-1000 0

10

100 200 300 Temperature rise [K]

400

Fig. 11. The variation of the first 5 natural frequencies of the FGM plate subjected to different uniform temperature rise.

3

4

10 Frequency [Hz]

10

Fig. 13. Sound radiation efficiency of FGM plate subjected to different temperature rise.

1240

1.4 Critical frequency [Hz]

Natural frequency ratio

1.2 1 0.8 0.6 0.4 0.2 0 0

Temperature dependent Temperature independent

1220

mode (1,1) mode (1,2) mode (2,1) mode (1,3) mode (3,1)

100 200 300 Temperature rise [K]

1200

1180

1160

1140 0

20

40 60 80 Temperature rise [K]

100

400 Fig. 14. The critical frequency of FGM plate subjected to different temperature rise with and without considering temperature dependent material properties.

Fig. 12. The variation of the first 5 natural frequency ratios of the FGM plate subjected to different uniform temperature rise.

plate (h = 0.01 m, N = 1) subjected to different temperature rise is studied and the results are presented in Fig. 13. The thermal field across the thickness is uniform and the temperature rise of DT = 0 K, DT = 200 K and DT = 400 K are considered. it can be seen from Fig. 13 that the radiation efficiency of the plate subjected to different temperature rise share a similar tendency. The radiation efficiency increase with the increase of frequency until approaches a maximum and then gradually tends to unity thereafter. There is no significant variation of sound radiation efficiency of FGM plate subjected to different temperature rise in the low-frequency range and the high frequency range, however, the radiation efficiency of the plate generally decreases with an increase of temperature rise in the short-circuiting region (the frequency range between the fundamental frequency and the critical frequency [40]). The critical frequency has been identified as an important parameter in vibro-acoustic response of plate structures. The maximum radiation efficiency occurs generally around the critical frequency. The critical frequency is defined as the frequency where the wavelength of flexural waves in the plate equals the wavelength of acoustic waves in the fluid, generally air [36]

fc ¼

c20 2p

rffiffiffiffiffiffi mS D

ð41Þ

Where ms and D are the surface density and the bending stiffness of the plate, respectively, and they are defined as

Z mS ¼

0:5h

Z D¼

0:5h

0:5h

0:5h

qðz; TÞdz Eðz; TÞ

1  mðz; TÞ2

ð42Þ

z2 dz

ð43Þ

For an homogeneous isotropic plate, the critical frequency in
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq. (41) is f c ¼ c20 12qð1  m2 Þ=E =2ph. Fig. 14 shows the variation of the critical frequency of a FGM plate (h = 0.01 m, N = 1) with the increase of temperature rise. It can be seen that the critical frequency increase with the increase of temperature rise when considering the temperature dependent material properties, this is mainly because the effective elastic modulus decrease with the increase of temperature rise. However, the critical frequency stays the same when the temperature dependent material properties are not taken into account. When the temperature rise is significant, the difference between the critical frequency of FGM plate with and without considering the temperature dependent material properties is considerable.

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T. Yang et al. / Composite Structures 144 (2016) 165–176

5. Conclusions The vibro-acoustic characteristics of functionally graded materials plates in thermal environment are presented in this paper. The material properties are supposed to be temperature dependent and vary continuously along the thickness direction according to power law distribution. The temperature is constant in the plane of the plate and varies only in the thickness direction. The first-order shear deformation plate theory is obtained to derive the governing equations of the FGM plate subjected to thermal environment through Hamilton’s principle. The acoustic response of the FGM plates subjected to point load is calculated with Rayleigh integral. Several validation studies are carried out to verify the accuracy of the present formulation both for the structural and the acoustic model, and good agreements are achieved. Finally, some parametric studies on the effect of temperature dependent material properties, material distribution, temperature fields and temperature rise are carried out to investigate the acoustic response characteristics of the FGM plates in thermal environment. Following the investigations, several observations are made with respect to the vibro-acoustic response of the FGM plates in thermal environment. The temperature dependent material properties have significant effects on the vibro-acoustic response of the FGM plates in thermal environment. The peaks of the sound power level shift to higher frequency domain due to overestimating the natural frequencies when temperature dependent material properties are not considered. With the temperature rise increase, the discrepancy between the sound radiation results of FGM plates with and without considering the temperature dependent material properties increase dramatically. The peaks of sound power level shift towards lower frequency range with the increase of the power-law index. The temperature variation through the thickness direction also plays an important role in the vibro-acoustic response of the FGM plates. There is no considerable discrepancy between the results of the FGM plates subjected to the linear temperature rise and the nonlinear temperature rise. However, for the same temperature rise, there is a considerable discrepancy between the results of sound radiation of the FGM plates subjected to the uniform temperature rise and linear temperature rise or the nonlinear temperature rise. Finally, it is found that peaks of the sound power level shift to low frequency domain with the increasing the temperature rise, and the first natural frequency is much more sensitive to the temperature change than the other natural frequencies. Several practical applications are related to this analytical formulations developed in this study such as vibro-acoustic sensitivity analysis, the material phase distribution identification and parameter optimization. Acknowledgments This research is supported by the National Natural Science Foundation of China (Nos. 51575201, 11204098 and 51405093). The authors gratefully acknowledge all of these supports. References [1] Jha DK, Kant T, Singh RK. A critical review of recent research on functionally graded plates. Compos Struct 2013;96:833–49. [2] Gupta A, Talha M. Recent development in modeling and analysis of functionally graded materials and structures. Prog Aerosp Sci 2015. [3] Thai H-T, Kim S-E. A review of theories for the modeling and analysis of functionally graded plates and shells. Compos Struct 2015;128:70–86. [4] Koizumi M. FGM activities in Japan. Compos B Eng 1997;28:1–4. [5] Cao XT, Hua HX, Zhang ZY. Sound radiation from shear deformable stiffened laminated plates. J Sound Vib 2011;330:4047–63. [6] Täger O, Dannemann M, Hufenbach WA. Analytical study of the structuraldynamics and sound radiation of anisotropic multilayered fibre-reinforced composites. J Sound Vib 2015;342:57–74.

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