Generalized thermoelastic waves in functionally graded plates without energy dissipation

Composite Structures 93 (2010) 32–39

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Generalized thermoelastic waves in functionally graded plates without energy dissipation Yu Jiangong *, Zhang Xiaoming, Xue Tonglong School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454003, PR China

a r t i c l e

i n f o

Article history: Available online 26 June 2010 Keywords: Functionally graded materials Thermoelastic waves Without energy dissipation Legendre polynomial Dispersion curves

a b s t r a c t In this paper, a dynamic solution of the propagating thermoelastic waves in functionally graded material (FGM) plate subjected to stress-free, isothermal boundary conditions is presented in the context of the Green–Naghdi (GN) generalized thermoelastic theory. The FGM plate is composed of two orthotropic materials. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The coupled wave equation and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of the inhomogeneous thermoelastic plate and the corresponding pure elastic plate are compared to show the characteristics of thermal modes and the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions of elastic modes and thermal modes are shown to discuss their differences. A plate with a different gradient variation is calculated to illustrate the influence of the gradient field on the wave characteristics. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Over the past few years, functionally graded materials (FGM) have received much attention with their increasing usage in various applications and working environments because of their gradually changed material properties. With the purpose of the non-destructive evaluation, the wave characteristics in FGM structures have attracted considerable research efforts. Liu et al. developed a finite layer element method to investigate the Lamb waves in the FGM [1] and functionally graded piezoelectric materials (FGPM) [2] plates. Ohyoshi [3] improved each finite layer element into linearly inhomogeneous layer element (LIE), i.e. using the piecewise linear interpolation approximating the continuous gradient variation. Liu et al. [4] gave the general solution of LIE model, and obtained the transient response of FGM plate through the Fourier transformation. Based on the LIE method and Hamilton principle, Han et al. obtained the dispersion relations of the FGM [5] and FGPM [6] hollow cylinders. Han et al. [7] also proposed a quadratic layer element method to investigate the wave in FGM plates. Based on the first-order shear deformation theory, Chakraborty et al. [8] developed a new beam element to study the wave propagation in FGM beam structures. Chakraborty and Gopalakrishnan developed the gradient spectral elements to investigate the wave propagation in the FGM beam [9,10]. Wang and Rokhlin [11] used the recursive geometric integrators for the wave propagation in multilayered graded elastic medium. Chen et al. [12] calculated the dispersion * Corresponding author. Tel.: +86 1 369 391 965 1; fax: +86 3 913 983 207. E-mail address: [email protected] (J. Yu). 0263-8223/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2010.06.020

curves through reverberation matrix method. Shakeri et al. [13] studied the vibration and radial wave propagation in FGM thick hollow cylinder that is made of many isotropic sub-cylinders. Hosseini [14] obtained the axisymmetrical dynamic solution of an isotropic layered FGM thermoelastic hollow cylinder (without energy dissipation) by Galerkin finite element and Newmak methods. All the above research having a common point, they divided the FGM structure into many homogeneous or inhomogeneous layers. In some other work, the FGM is taken as continuous gradient medium. For some continuous graded mediums with simple gradient fields, their analytical solutions have been obtained. Abd-Alla and Ahmed obtained the analytical solutions of the Love waves [15], Stoneley and Rayleigh waves [16] in inhomogeneous orthotropic elastic medium. Du et al. [17] obtained the analytical solutions of the Love wave in FGPM layer. Ma et al. [18] studied the anti-plane shear wave scattering by a Griffth crack in continuously exponential function varying FGMP medium. The gradient variations studied by the above four literatures are all exponential gradient fields. Eskandari and Shodja [19] obtained the analytical solutions of Love wave propagation in FGPM with quadratic gradient variation. For more complicated continuous graded structures, researchers resorted to the approximate or asymptotic methods. Using the WKB method, Li et al. [20] and Liu and Wang [21] investigated the propagation of Love wave in FGPM structures. Lefebvre et al. [22] developed the Legendre orthogonal polynomial series method to investigate the wave propagation in FGPM plates, and then in FGM hollow cylinders by Elmaimouni et al. [23]. Yu et al. using this polynomial method investigated the wave characteristics in FGM [24], FGPM [25] spherical curved plates and FGPM cylindrical curved plates [26], FGPM

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J. Yu et al. / Composite Structures 93 (2010) 32–39

hollow cylinders [27]. Wu et al. extended this method to the wave propagation in the magneto-electro-elastic FGM plates [28]. Then Yu et al. using this method investigated the magneto-electro-elastic FGM hollow cylinders [29] and cylindrical curved plates [30]. From the above simple review we can see that the wave propagation in FGM structures has received much attention, and that the correspondingly coupled electro-elastic and magneto-electroelastic FGM structures also obtained appropriate researching effort. At present, the FGM are mainly used in the heat insulation structures. But few literatures paid attention to the thermoelastic wave propagation in FGM structures. As far as authors know, only Chakraborty et al. [8–10] and Hosseini [14] considered this topic using finite element method. In this article, the propagation of thermoelastic waves in continuously orthotropic FGM plate is investigated in the context of the Green–Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The dispersion curves, displacement distributions, temperature distributions and stress distributions are illustrated. The influences of the gradient variation on the wave characteristics are discussed. In this paper, the stress-free, isothermal boundary conditions are assumed.

Here Ki are the material constant characteristics of the theory (not the coefficients of thermal conductivity) and Ce is the specific heat at constant strain. By introducing the rectangular window function p(z)

pðzÞ ¼



1;

06z6h

0;

elsewhere

the stress-free boundary conditions (Tzz = Txz = Tyz = 0 at z = 0, z = h) are automatically incorporated in the constitutive relations:

T xx ¼ C 11 exx þ C 12 eyy þ C 13 ezz  b1 T T yy ¼ C 12 exx þ C 22 eyy þ C 23 ezz  b2 T T zz ¼ ðC 13 exx þ C 23 eyy þ C 33 ezz  b3 TÞpðzÞ

ð4Þ

T yz ¼ 2C 44 eyz pðzÞ T xz ¼ 2C 55 exz pðzÞ T xy ¼ 2C 66 exy

Because the material properties vary in the thickness direction, the elastic constants of the medium are the function of z

 z 1

 z 2

 z L

2. Theoretical formulation

C ij ðzÞ ¼ C ij þ C ij

Consider an orthotropic, thermoelastic FGM plate which is infinite horizontally with a thickness h. We place the horizontal (x, y)plane of a cartesian coordinate system on the bottom surface and let the plate be in the positive z-region. The orthotropic thermoelastic constitutive equation can be written in the following form:

With implicit summation over repeated indices, Cij(z) can be written compactly as

ð0Þ

T xx ¼ C 11 exx þ C 12 eyy þ C 13 ezz  b1 T T yy ¼ C 12 exx þ C 22 eyy þ C 23 ezz  b2 T ð1Þ

T xz ¼ 2C 55 exz T xy ¼ 2C 66 exy where Tij, eij are the stress and strain; Cij, bi are the elastic and volume expanding coefficients; T is the temperature change above the uniform reference temperature ‘‘T0”. The relationship between the strain and displacement can be expressed as

eyz exy

@ux @uy @uz ¼ ; eyy ¼ ; ezz ¼ ; @x @y @z     1 @uy @uz 1 @ux @uz ¼ þ ; exz ¼ þ ; 2 @z 2 @z @y @x   1 @ux @uy ¼ þ 2 @y @x

ð2Þ

where ui are the elastic displacements. The dynamic equation for the thermoelastic plate in the absence of body forces and heat sources is governed by 2

@T xx @T xy @T xz @ ux þ þ ¼q 2 @x @y @z @t

 z l h

ðlÞ

; bi ðzÞ ¼ bi h l ¼ 0; 1; 2; . . . ; L

ðLÞ

þ    þ C ij

h

 z l h

;

ðlÞ

K i ðzÞ ¼ K i

 z l h

; ð5Þ

For a free harmonic wave being propagated in the x direction in the plate, we assume the displacement components and temperature change to be of the form

ux ðx; y; z; tÞ ¼ expðikx  ixtÞUðzÞ

ð6aÞ

uy ðx; y; z; tÞ ¼ expðikx  ixtÞVðzÞ

ð6bÞ

uz ðx; y; z; tÞ ¼ expðikx  ixtÞWðzÞ

ð6cÞ

Tðx; y; z; tÞ ¼ expðikx  ixtÞXðzÞ

ð6dÞ

U(z), V(z), W(z) represent the amplitude of vibration in the x, y, z directions respectively and X(z) represents the amplitude of temperature change. k is the magnitude of the wave vector in the propagation direction, and x is the angular frequency. Substituting Eqs. (2), (4)–(6) into Eq. (3), the governing differential equations in terms of displacement components and temperature change can be obtained:

 zl h ðlÞ 00 1 ðlÞ 0 ðlÞ 1 ðlÞ 0 C U þ lz C U þ ikC W þ likz C W pðzÞ 55 55 55 55 l h i 2

ðlÞ

ðlÞ

þ ðdðz  0Þ  dðz  hÞÞ ¼

!

2

¼ qC e

h

l ¼ 0; 1; 2; . . . ; L

 z l

2

@2T @2T @2T @ 3 ux @ 3 uy @ 3 uz K 1 2 þ K 2 2 þ K 3 2  T 0 b1 þ b2 þ b3 2 2 @x @y @z @x@t @y@t @z@t 2

ð2Þ

þ C ij

ðlÞ

þ ikC 13 W 0  k C 11 U  ikb1 X

@T xy @T yy @T yz @ 2 uy þ þ ¼q 2 @x @y @z @t @T xz @T yz @T zz @ uz þ þ ¼q 2 @x @y @z @t

h

Other material constants can be treated in the same way,

qðzÞ ¼ qðlÞ

T zz ¼ C 13 exx þ C 23 eyy þ C 33 ezz  b3 T T yz ¼ 2C 44 eyz

exx

ðlÞ

C ij ðzÞ ¼ C ij

ð1Þ

@ T @t 2 ð3Þ

qðlÞ zl x2 l

h

zl l

h

ðlÞ

C 55 ðU 0 þ ikWÞ

U

i  zl h ðlÞ 00 1 ðlÞ 0 2 ðlÞ C V þ lz C V p ðzÞ  k C V 44 44 66 l h zl ðlÞ qðlÞ zl x2 V þ ðdðz  0Þ  dðz  hÞÞ l C 44 V 0 ¼  l h h

ð7aÞ

ð7bÞ

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J. Yu et al. / Composite Structures 93 (2010) 32–39

zl n h

ðlÞ

1

ðlÞ

1

ðlÞ

ðlÞ

1

ikC 13 U 0 þ lz C 13 U þ C 33 ðW 00 þ lz W 0 Þ  b3 ðX 0 þ lz XÞ h o ðlÞ 2 ðlÞ þ ikC 55 U 0  k C 55 W þ ðdðz  0Þ  dðz  hÞÞ  zl  ðlÞ qðlÞ zl x2 ðlÞ ðlÞ  l C 33 W 0 þ ikC 13 U  b3 X ¼  W l h h l

 zl   z  ðlÞ 00 2 ðlÞ ðlÞ ðlÞ K 3 X  k K 1 X þ l T 0 ikb1 x2 U þ b3 x2 W 0 l h h ! L X zl ðlÞ zl ðlÞ 2 C q xX ¼ l e l h l¼0 h

i

pðzÞ

XðzÞ ¼ zðz  hÞ

!

zl

ðlÞ

qðlÞ x2 U þ T 0 l

b l 1 h

!

L X zl

ðlÞ

b l 3

h l¼0 h !   zl ðlÞ zl 2 ðlÞ ðlÞ 00 þ q K T  k K T 3 1 l l h l¼0 h ! ! L L X X zl ðlÞ zl ðlÞ zl ðlÞ 2 q C q xT ¼ l l e l h l¼0 h l¼0 h l¼0

zl h

l

p3m Q m ðzÞ

ð11Þ

ð7cÞ Table 1 The material properties of the two materials.

ð7dÞ

Property

C11

C13

C33

C55

b1

b3

Ce

q

Si3N4 Zinc

574 162.8

127 50.8

433 62.7

108 38.5

3.22 3.85

2.71 5.07

670 390

3.2 7.14

Units: Cij (109 N m2), bi (106 N deg1 m2), Ce (J Kg deg m1), q (103 kg m3).

Here, Eq. (7b) is independent of the other three equations. In fact, Eq. (7b) represents the propagating SH waves. Eqs. (7a) and (7c) control the propagating Lamb-like waves. They are coupled with the heat conduction equation. The SH waves in pure elastic FGM plates have received much attention. In this paper, only the coupled thermoelastic Lamb-like waves are considered. Eq. (7d) can be written as: L X zl

1 X m¼0

l

ikT 0

and expand X(z) to

(a)

(104rad/s) 3

2

qðlÞ x2 W 0

L X

1

ð8Þ

Here

zl h

l

M=8 M=7 M=6 M=5 1

qðlÞ x2 W 0 ¼



zl l

h

qðlÞ x2 W

0

 lz

1

l

zl h

l

(b)

qðlÞ x2 W

2

3

4

5

6

7

8

kh

(104rad/s)

l

So, we can eliminate z l qðlÞ x2 U and z l qðlÞ x2 W 0 through Eqs. (7a) h h and (7c). Then Eq. (7d) becomes

!  zl h ðlÞ  00 1 2 ðlÞ ðlÞ ðlÞ  ikT 0 b C 55 U þ lz U  k C 11 U þ ikC 13 W 0 l 1 l h h l¼0 i ðlÞ 1 ðlÞ þ ikC 55 ðW 0 þ lz WÞ  ikb1 X pðzÞ þ ðdðz  0Þ  dðz  hÞÞ !  L X zl ðlÞ zl ðlÞ zl h ðlÞ 0 ðlÞ b ikC 13 U þ ikC 55 U 0  l C 55 ðU 0 þ ikWÞ  T 0 l 3 l h h h l¼0  i 1 ðlÞ 1 2 ðlÞ 1 ðlÞ ðlÞ þ lz C 13 U þ C 33 ðW 00 þ lz W 0 Þ  k C 55 W  b3 X 0 þ lz X pðzÞ  0 zl  ðlÞ ðlÞ ðlÞ þ dðz  0Þ  dðz  hÞÞ l C 33 W 0 þ ikC 13 U  b3 X ! h L X zl ðlÞ zl h ðlÞ 0 1 ðlÞ 1 ðlÞ þ lz T 0 b ikC 13 U þ ikC 55 U 0 þ lz C 13 U l 3 l h h l¼0  i 1 2 ðlÞ 1 ðlÞ ðlÞ þ C 33 ðW 00 þ lz W 0 Þ  k C 55 W  b3 X 0 þ lz X pðzÞ   zl  ðlÞ ðlÞ ðlÞ þ dðz  0Þ  dðz  hÞÞ l C 33 W 0 þ ikC 13 U  b3 X h ! L  l l  X z ðlÞ z 2 ðlÞ ðlÞ þ q K 3 T 00  k K 1 T l l h l¼0 h ! ! L L l X X z ðlÞ zl ðlÞ zl ðlÞ 2 ¼ q C q xT ð9Þ l l e l h l¼0 h l¼0 h

4

L X zl

To obtain the solution of the thermoelastic Lamb-like waves controlled by the coupled Eqs. (7a), (7c) and (9), we expand U(z) and W(z) to Legendre orthogonal polynomial series

UðzÞ ¼

1 X m¼0

p1m Q m ðzÞ;

WðzÞ ¼

1 X m¼0

p2m Q m ðzÞ

ð10Þ

3

M=8 M=7 M=6 M=5

2

1

(c)

2

3

4

5

6

7

kh

8

(104rad/s) 6

5

4 M=8 M=7 M=6 M=5 1

2

3

4

5

6

7

kh

8

Fig. 1. Dispersion curves for the FGM thermoelastic plate for various ‘‘M”: (a) the first four modes; (b) modes 5–8; (c) modes 9–12.

35

J. Yu et al. / Composite Structures 93 (2010) 32–39

Obviously, Eq. (11) automatically satisfies the isothermal boundary conditions (T = 0 at z = 0, z = h). Here pim ði ¼ 1; 2; 3Þ is the expansion coefficients and

Q m ðzÞ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2m þ 1 2z  h Pm h h

with Pm being the mth Legendre polynomial. Theoretically, m runs from 0 to 1. In practice, the summation over the polynomials in Eqs. (10) and (11) can be halted at some finite value m = M, when higher order terms become essentially negligible. Eqs. (7a), (7c) and (9) are multiplied by Qj(z) with j running from 0 to M. Then integrating over z from 0 to h gives the following 3(M + 1) equations l

1 l j;m 2 l j;m 3 2 l j 1 Aj;m 11 pm þ A12 pm þ A13 pm ¼ x  M m pm

ð12aÞ

l

1 l j;m 2 l j;m 3 2 l j 2 Aj;m 21 pm þ A22 pm þ A23 pm ¼ x  M m pm

ð12bÞ

l

j 1 l j;m 2 l j;m 3 2 l 3 Aj;m 31 pm þ A32 pm þ A33 pm ¼ x  MT m pm

ð12cÞ

where summation over repeated index mis implied with mranging j l from 0 to M; l Aj;m ab ða; b ¼ 1; 2; 3Þ and M m are the elements of a non-symmetric matrix. They can be obtained according to Eqs. (7), (9) and are given in Appendix A. Eq. (12) can be written as:

2l

Aj;m 11 6 6 l j;m 6 A21 4 l

Aj;m 31

l

Aj;m 12

l

Aj;m 22

l

Aj;m 32

38 9 2l j 1 Aj;m Mm 13 > pm > < > = 7> 6 26 2 l j;m 7 A22 7 pm ¼ x 6 0 5> 4 > > > : ; 3 l j;m p 0 A33 m l

0 l

Mjm 0

38 9 p1 > > m> > = 7< 7 0 7 p2m > 5> > : 3 > ; j l p MT m m

Based on the foregoing formulation, a computer program has been written to calculate the dispersion curves and displacement, temperature, stress distributions for the FGM thermoelastic plate composed of Si3N4 (bottom) and zinc (top) with thickness h = 1 m. Their material constants are listed in Table 1 except for Ki , the material constant characteristics of the theory. We take their values as K1 = CeC11/4 and K3 = CeC33/4 [31]. The gradient field used here is

CðzÞ ¼ C S þ ðC zinc  C S Þðz=hÞ

ð16Þ

3.1. Convergence of the problem In order to observe the convergence of the method, the thermoelastic wave dispersion curves are calculated when M = 5, 6, 7 and 8 respectively, as shown in Fig. 1. Fig. 1a shows the dispersion curves of first four modes, Fig. 1b being the modes 5–8 and Fig. 1c being the modes 9–12. It can be seen that for the first five modes, the dispersion curves are coincident for M = 5, 6, 7 and 8, which indicates that the first five modes are convergent when M = 5. Similarly, it can be concluded that the first six modes are convergent when M = 6 and that the first eight modes are convergent when M = 7. So, we assume that at least the first M modes are convergent.

0

(104rad/s) 5

ð13Þ So, Eq. (13) yields a form of the eigenvalue problem. The eigenvalue x2 gives the angular frequency of the thermoelastic wave; eigenvectors pim ði ¼ 1; 2; 3Þ allows the components of the particle displacement and temperature change. The phase velocity can be obtained according to Vph = x/k. The complex matrix Eq. (13) can be solved numerically making use of standard computer programs for the diagonalization of non-symmetric square matrices. The matrix dimensionality in both members of Eq. (13) is (3M + 3)  (3M + 3). So 3(M + 1) eigenmodes are generated from the order M of the expansion. The solutions to be accepted are those eigenmodes for which convergence is obtained as M is increased. We determine that the eigenvalues obtained are converged solutions when a further increase in the matrix dimension does not result in a significant change in the eigenvalue. Calculation of stresses is a straightforward subsequent step: after evaluating the displacement and temperature, the stress distributions can be immediately calculated from Eqs. (1) and (2). The computer program was written using Mathmatica.

4

3

2

1

1

3. Numerical results

12

In this paper, the Voigt-type model is used to calculate the effective modulli of the FGM. It is expressed as

10

ð14Þ

where Vi(z) and Pi respectively denote the volume fraction of the ith material and the corresponding property of the ith material. Here, P V i ðzÞ ¼ 1. So, the properties of the graded material can be expressed as

PðzÞ ¼ P2 þ ðP1  P2 ÞV 1 ðzÞ

3

4

5

6

7

8

kh

Fig. 2. Frequency spectra for FGM plate: solid line, pure elastic plate; dotted line, thermoelastic plate.

14

PðzÞ ¼ P1 V 1 ðzÞ þ P2 V 2 ðzÞ

2

Vph (Km/s)

8 6 4 2

ð15Þ

According to Eq. (5), the gradient variation of the material volume fraction can be expressed as a power series expansion. The coefficients of the power series can be determined using the Mathematica function ‘Fit’.

1

2

3

4

5

6

7

8

f (KHz) Fig. 3. Phase velocity spectra for FGM plate: solid line, pure elastic plate; dotted line, thermoelastic plate.

36

J. Yu et al. / Composite Structures 93 (2010) 32–39

×10-5 arbitrary unit 6

ux

10

×10-7 arbitrary unit

×10-5 arbitrary unit

15

10

4

5

5

ux

2

0

0

-5

-2

uz

-10

-5

uz

-4

-15

uz

ux

0

-10

-6

0

0.2

0.4

0.6

0.8

1

0

0.2

Thickness coordinate (z)(m) Mode 4

0.4

4

1

0

uz

0.6

0.8

1

0

uz

-5

ux

-1

-4

0.4

ux

5

0

uz

0.2

Thickness coordinate (z)(m) Mode 6 ×10-7 arbitrary unit

ux

-2

0

1

×10-5 arbitrary unit

×10-5 arbitrary unit

2

0.8

0.6

Thickness coordinate (z)(m) Mode 5

-10 0

0.2

0.4

0.6

0.8

1

0

0.2

Thickness coordinate (z)(m) Mode 7

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Thickness coordinate (z)(m) Mode 9

Thickness coordinate (z)(m) Mode 8

Fig. 4. Displacement distributions of the FGM thermoelastic plate at kh = 1.5.

×10-1 arbitrary unit

×10-1 arbitrary unit

×10-1 arbitrary unit

3

0

2

2

1

-1 -2 -3 0

0.2

0.4

0.6

0.8

1

0

0

-1

1

0

Thickness coordinate (z)(m) Mode 4

0.2

0.4

0.6

0.8

0

1

×10-1 arbitrary unit

×10-1 arbitrary unit

1 0 -1 -2

0.2

0.4

0.6

0.8

1

Thickness coordinate (z)(m) Mode 6

Thickness coordinate (z)(m) Mode 5

×10-1 arbitrary unit

1

1

0

0

-1

-1

-2 0

0.2

0.4

0.6

0.8

1

0

Thickness coordinate (z)(m) Mode 7

0.2

0.4

0.6

0.8

Thickness coordinate (z)(m) Mode 8

1

0

0.2

0.4

0.6

0.8

1

Thickness coordinate (z)(m) Mode 9

Fig. 5. Temperature change distributions of the FGM thermoelastic plate at kh = 1.5.

For all the undermentioned calculations, the series expansions Eqs. (9) and (10) are truncated at M = 10. 3.2. Wave characteristics in the FGM thermoelastic plate The thermoelastic wave frequency spectra and phase velocity spectra for the FGM plate are shown in Figs. 2 and 3, in which

the solid lines are the dispersion curves of the pure elastic plate and the dotted lines are those of the thermoelastic plate. It can be seen that the dispersion curves of some modes in the thermoelastic plate are almost coincident with the dispersion curves in the pure elastic plate. Obviously, these modes are elastic modes and they are little influenced by the thermoelasticity. The other modes in the thermoelastic plate are thermal modes. They do not

37

J. Yu et al. / Composite Structures 93 (2010) 32–39

×107 arbitrary unit

×107 arbitrary unit 4

×105 arbitrary unit 8

Tzz

Txz

-1

0

Tzz

4

-4

-3

Txz

0

Txx

Txx

-8

Txz

Tzz

Txx

-4

-5

0

0.2

0.6

0.4

0.8

0

1

Thickness coordinate (z)(m) Mode 4

0.4

0.2

0.8

1

0

1

0.4

0.6

0.8

1

×105 arbitrary unit

Tzz

Tzz

0.2

Thickness coordinate (z)(m) Mode 6

×107 arbitrary unit

×107 arbitrary unit 2

0.6

Thickness coordinate (z)(m) Mode 5

Txz

Txx

4

0.5

0

0

0

Txz

-2

Txx 0

Txx

-4

-0.5 0.2

0.4

0.6

0.8

0

1

0.2

Tzz

0.4

0.6

0.8

1

Txz 0

Thickness coordinate (z)(m) Mode 8

Thickness coordinate (z)(m) Mode 7

0.2

0.4

0.6

0.8

1

Thickness coordinate (z)(m) Mode 9

Fig. 6. Stress distributions of the FGM thermoelastic plate at kh = 1.5.

Txz and the temperature change T are zero. In addition, more attention should be paid to the fact that the displacement and stress amplitudes of thermal modes are far less than those of elastic modes, but the temperature change amplitudes of thermal modes and elastic modes are similar.

Vph (Km/s) 14 12 10

3.3. The influence of the gradient field

8

4

In this section, another FGM thermoelastic plate is studied. Its material and size is the same to the first example, but the gradient field is changed to quadratic variation,

2

CðzÞ ¼ C S þ ðC zinc  C S Þðz=hÞ2

6

1

2

3

4

5

6

7

8

f (KHz) Fig. 7. Phase velocity spectra for the quadratic gradient FGM plate: solid line, pure elastic plate; dotted line, thermoelastic plate.

exist in the pure elastic plate. According to Eqs. (7a) and (7c), the value of elastic modes is determined by the elastic constants Cij and volume expanding coefficients bi. The influence of thermoelasticity on elastic modes comes from the volume expanding coefficients b1 and b3. From Table 1, the elastic constants are about ten thousand times of the volume expanding coefficients, which leads to the influence of thermoelasticity being little. From the dispersion curves, those thermal modes have similar behavior as the elastic modes. Their phase velocities decrease with the increasing of frequency, and eventually congregate together with the elastic modes. Figs. 4–6 show the displacement distributions, temperature change distributions and stress distributions of the modes 4–9 at kh = 1.5, of which modes 6 and 9 are thermal modes and the other four modes are elastic modes. It can be checked on these figures that on the upper and bottom surfaces of the plate, the stress Tzz,

ð17Þ

Fig. 7 gives its phase velocity dispersion curves. In comparison with Fig. 3, we find that the velocity values of all the elastic modes and the thermal modes in the quadratic FGM plate are higher than those in the linear FGM plate. Table 1 indicates that the wave velocity in Si3N4 is higher than that in zinc. According to Eqs. (16) and (17), the volume fraction of Si3N4 in quadratic FGM plate is larger than that in quadratic FGM plate. 4. Conclusions In the context of the generalized GN thermoelastic theory, the wave characteristics in the FGM plate are discussed. Based on the calculated results, the following conclusions can be drawn: (a) For the wave propagation in the orthotropic FGM thermoelastic plate, the independent SH wave is not influenced by the thermoelasticity. (b) For the propagating thermoelastic waves in the FGM plate, the thermoelasticity has little influence on the elastic modes while the displacements and stresses generated by thermal modes are far less than those by elastic modes. (c) The gradient field has obvious influence both on elastic modes and on thermal modes.

38

J. Yu et al. / Composite Structures 93 (2010) 32–39

Acknowledgments

l

The work is supported by the National Natural Science Foundation of China (No. 10802027) and by the high-performance grid computing platform of Henan Polytechnic University. The authors also wish to express their sincere thanks to the reviewers for their careful comments.

l

Aj;m 11

Aj;m 12

l

¼h

h

MT jm

¼ h

l

b

Q j ðzÞQ m ðzÞdz

a

Z

C ðlÞ e

b

Q j ðzÞQ m ðzÞzðz  hÞ

a

ðlÞ lC 55 Aðm; j; l

 1; 1Þ i 2 ðlÞ ðlÞ  k C 11 Aðm; j; l; 0Þ þ C 55 Bðm; j; l; 1Þ h  l ðlÞ ðlÞ ðlÞ ¼ ikh C 13 þ C 55 Aðm; j; l; 1Þ þ lC 55 Aðm; j; l  1; 0Þ i ðlÞ þ C 55 Bðm; j; l; 0Þ þ

Z

l ðlÞ

Aj;m 13 ¼ ikh b1 AAðm; j; l; 0Þ l

h

n

ðlÞ

Aj;m 21 ¼ ikh

l

Aj;m 22 ¼ h

l

k l

l

l

2

Bðm; j; i; nÞ ¼

Z

@ n Q m ðzÞ @ðHðzÞ  Hðz  hÞÞ dz @zn @z

a

Z

AAðm; j; i; nÞ ¼

Q j ðzÞziþ1 ðz  hÞ

@ n Q m ðzÞ dz @zn

Q j ðzÞziþ1 ðz  hÞ

@ n Q m ðzÞ @ðHðzÞ  Hðz  hÞÞ dz @zn @z

b

a

BBðm; j; i; nÞ ¼

Z

b

þ

l ðlÞ

Aj;m 32

l

h

AAT3ðm; j; i; nÞ ¼

C 55 AT1ðm; j; l; 2Þ þ lC 55 AT1ðm; j; l  1; 1Þ i 2 ðlÞ ðlÞ  k C 11 AT1ðm; j; l; 0Þ þ C 55 BT1ðm; j; l; 1Þ h  l ðlÞ ðlÞ ðlÞ  ikT 0 h C 13 þ C 55 AT3ðm; j; l; 2Þ þ lC 13 AT3ðm; j; l  1; 1Þ   ðlÞ ðlÞ ðlÞ þ 2C 13 þ C 55 BT3ðm; j; l; 1Þ þ C 13 lðl  1ÞAT3ðm; j; l  2; 0Þ i ðlÞ ðlÞ þ C 13 lBT3ðm; j; l  1; 0Þ þ C 13 BT3ðm; j; l; 0Þ h  l ðlÞ ðlÞ þ ikT 0 h C 13 þ C 55 ATMðm; j; l; 1Þ i ðlÞ ðlÞ þ lC 13 ATMðm; j; l  1; 0Þ þ C 13 BTMðm; j; l; 0Þ h  2 l ðlÞ ðlÞ ðlÞ ¼ k T0h C 13 þ C 55 AT1ðm; j; l; 1Þ þ lC 55 AT1ðm; j; l  1; 0Þ i h l ðlÞ ðlÞ þ C 55 BT1ðm; j; l; 0Þ  T 0 h C 33 AT3ðm; j; l; 3Þ 2

 þ

ðlÞ C 33 BTT3ðm; j; l; 0Þ

i

k

2

ðlÞ C 55 ATMðm; j; l; 0Þ

ðlÞ

þ C 33 BTMðm; j; l; 1Þ l

Aj;m 33 ¼ h

h

þh

ðlÞ

i

þ

l

h

2

Q j ðzÞz

iþ1

! L @ n Q m ðzÞ X zl ðlÞ ðz  hÞ b dz l 3 @zn l¼0 h

BT1ðm;j;i;nÞ ¼

Z

Q j ðzÞzi

a

BT3ðm; j;i;nÞ ¼

Z

b

ðlÞ

þ lAAT3ðm; j; l  1; 1Þ þ 2BBT3ðm; j; l; 1Þ þ lðl  1ÞAAT3ðm; j; l  2; 0Þ þ lBBT3ðm; j; l  1; 0Þ þ lAATMðm; j; l  1; 0Þ þ BBTMðm; j; l; 0Þ

i@

n

Q j ðzÞz a

Z

BBT3ðm; j; i; nÞ ¼

b

Q j ðzÞziþ1 ðz  hÞ

a

@ n Q m ðzÞ @zn

! L @ðHðzÞ  Hðz  hÞÞ X zl ðlÞ dz b l 3 @z l¼0 h

 Z

! L Q m ðzÞ @ðHðzÞ  Hðz  hÞÞ X zl ðlÞ b dz l 3 @zn @z l¼0 h

b

Q j ðzÞz

i@

n

! L Q m ðzÞ @ 2 ðHðzÞ  Hðz  hÞÞ X zl ðlÞ b dz l 3 @zn @z2 l¼0 h

2; 1Þ

K 3 AAMðm; j; l; 2Þ  k K 1 AAMðm; j; l; 0Þ i 2 ðlÞ ðlÞ l  k b1 T 0 AAT1ðm; j; l; 0Þ þ T 0 b3 h ½AAT3ðm; j; l; 2Þ

ðlÞ l

! L @ n Q m ðzÞ @ðHðzÞ  Hðz  hÞÞ X zl ðlÞ dz b l 1 @zn @z l¼0 h

b

BBTT3ðm; j; i; nÞ ¼

Z

b

Q j ðzÞziþ1 ðz  hÞ

a

@ n Q m ðzÞ @zn

! L @ 2 ðHðzÞ  Hðz  hÞÞ X zl ðlÞ  b dz l 3 @z2 l¼0 h

 1; 1Þ

þ BBTT3ðm; j; l; 0Þ  T 0 b3 h ½AATMðm; j; l; 1Þ

b

a

ðlÞ C 33 ATMðm; j; l; 2Þ

ðlÞ lC 33 ATMðm; j; l

! L @ n Q m ðzÞ X zl ðlÞ ðz  hÞ b dz l 1 @zn l¼0 h

a

BTT3ðm;j;i;nÞ ¼

ðlÞ

ðlÞ

ðlÞ ðlÞ 2C 33 BT3ðm; j; l; 2Þ þ lðl  1ÞC 33 AT3ðm; j; l  2 ðlÞ ðlÞ k C 55 BT3ðm; j; l; 0Þ þ lC 33 BT3ðm; j; l  1; 1Þ

þ

Z

ðlÞ

ðlÞ

Q j ðzÞz

iþ1

b

a

Aj;m 23 ¼ ikh b3 ½AAðm; j; l; 1Þ þ AAðm; j; l  1; 0Þ þ BBðm; j; l; 0Þ Aj;m 31 ¼ ikT 0 h

i

Z

AAT1ðm; j; i; nÞ ¼

 k C 55 AT3ðm; j; l; 1Þ þ lC 33 AT3ðm; j; l  1; 2Þ

l

! L @ n Q m ðzÞ X zl ðlÞ Q j ðzÞz b dz AT1ðm; j; i; nÞ ¼ l 1 @zn a l¼0 h ! Z b L @ n Q m ðzÞ X zl ðlÞ dz AT3ðm; j; i; nÞ ¼ Q j ðzÞzi b l 3 @zn a l¼0 h b

ðlÞ

ðlÞ C 33 Bðm; j; l; 1Þ

dz

Q j ðzÞzi

b

C 33 Aðm; j; l; 2Þ þ lC 33 Aðm; j; l  1; 1Þ

ðlÞ C 55 Aðm; j; l; 0Þ

q

@ n Q m ðzÞ dz @zn

Z

 ðlÞ ðlÞ ðlÞ C 13 þ C 55 Aðm; j; l; 1Þ þ lC 13 Aðm; j; l  1; 0Þ i ðlÞ þ C 13 Bðm; j; l; 0Þ

l

h

!2 ðlÞ

Q j ðzÞzi

b

a l

l

l¼0

a

ðlÞ C 55 Aðm; j; l; 2Þ

L X zl

where

Aðm; j; i; nÞ ¼

Appendix A l

l

Z

l

M jm ¼ h qðlÞ

ATMðm;j;i;nÞ ¼

Z

b

i@

Q j ðzÞz

a

Z

n

! ð0Þ PL lzl1 ðlÞ L Q m ðzÞ X zl ðlÞ q þ l¼1 hl q b dz PL zl ðlÞ l 3 @zn l¼0 h l¼0 hl q

L @ n Q m ðzÞ @ðHðzÞ  Hðz  hÞÞ X zl ðlÞ Q j ðzÞz b BTMðm; j; i; nÞ ¼ n l 3 @z @z a l¼0 h P l1 qð0Þ þ Ll¼1 lzhl qðlÞ dz  PL zl ðlÞ b

i

l¼0 hl

q

!

J. Yu et al. / Composite Structures 93 (2010) 32–39

AATMðm; j; i; nÞ ¼

Z

b

Q j ðzÞziþ1 ðz  hÞ

a



BBTMðm; j; i; nÞ ¼

AAMðm; j; i; nÞ ¼

Z

PL

lzl1 l¼1 hl PL zl ðlÞ l¼0 hl

qð0Þ þ

L @ n Q m ðzÞ X zl ðlÞ b l 3 @zn l¼0 h

qðlÞ

q

!

dz

@ n Q m ðzÞ @ðHðzÞ  Hðz  hÞÞ @zn @z a ! ð0Þ PL lzl1 ðlÞ L l X z ðlÞ q þ l¼1 l q h  b dz PL zl ðlÞ l 3 l¼0 h l¼0 hl q

Z a

b

Q j ðzÞziþ1 ðz  hÞ

b

Q j ðzÞz

iþ1

! L @ n Q m ðzÞ X zl ðlÞ ðz  hÞ q dz l @zn l¼0 h

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