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Three-Dimensional Thermo-Electro-Elastic Field in a Circular Plate of Functional Graded Materials with Transverse Isotropy a

a

a

Peidong Li , Xiangyu Li & Tao Wang a

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, China Accepted author version posted online: 30 Jun 2014.Published online: 30 Jun 2015.

Click for updates To cite this article: Peidong Li, Xiangyu Li & Tao Wang (2015) Three-Dimensional Thermo-Electro-Elastic Field in a Circular Plate of Functional Graded Materials with Transverse Isotropy, Mechanics of Advanced Materials and Structures, 22:7, 537-547, DOI: 10.1080/15376494.2013.828810 To link to this article: http://dx.doi.org/10.1080/15376494.2013.828810

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Mechanics of Advanced Materials and Structures (2015) 22, 537–547 C Taylor & Francis Group, LLC Copyright ISSN: 1537-6494 print / 1537-6532 online DOI: 10.1080/15376494.2013.828810

Three-Dimensional Thermo-Electro-Elastic Field in a Circular Plate of Functional Graded Materials with Transverse Isotropy PEIDONG LI, XIANGYU LI, and TAO WANG School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, China

Downloaded by [X.Y. Li] at 03:09 26 February 2015

Received 21 January 2012; accepted 20 December 2012.

This article presents a set of axisymmetric solutions for a circular plate of functional graded materials with transverse isotropy subjected to uniform thermal loadings on the upper and lower surfaces. The temperature field in the plate is obtained as a prior, by integration. A direct displacement method is then employed to explicitly derive the thermo-electro-elastic field induced by the thermal loadings. Numerical calculations are carried out to validate the present results and to show the effect of material heterogeneity on the distributions of the coupled field. Some interesting phenomena have been observed. Keywords: thermo-electro-elastic field, functional graded materials, circular plate, axisymmetric solutions, transverse isotropy

1. Introduction Piezoelectric medium, due to its electro-mechanical coupling effect, has now been widely used in intelligent structures and systems, as sensor and actuator [1]. The concept of functionally graded material (FGM) was initially proposed by a group of Japanese scientists to cater to the requirement of aerospace and aviation industries [2–5]. Now, the concept of FGM has been successfully extended to piezoelectric medium, i.e., functionally graded piezoelectric material (FGPM), in scientific and industrial communities [6, 7]. Development of mechanical behavior of intelligent circular plates before 2001 was thoroughly reviewed by Ding and Chen [1]. In the last decade, this topic still attracted a lot of attention from scholars. For instance, Sekouri et al. [8], based on the Kirchhoff plate model, proposed an analytical approach for modeling of a circular plate containing distributed piezoelectric actuators. Later, Fox et al. [9] employed a structural mechanics approach and presented a model to predict the deflections of a circular plate with an annular piezoelectric actuator. Recently, the coupled electro-mechanic field in a FGPM circular plate was studied by Wang et al. [6], using Fourier–Bessel expansion techniques. Then, Li et al. [7], adopting a direct displacement method, presented three-dimensional (3D) analytical solutions for FGPM under bending and tension.

Address correspondence to Xiangyu Li, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China. E-mail: [email protected] Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/umcm.

Many intelligent systems need to be operated in a thermal environment [10, 11]. Hence, it is natural to study the thermally induced elastic and electric fields. The pioneer and fundamental works on piezo-thermo-elastic behaviors are conducted by Mindlin [12, 13], Tiersten [14], Chandrasekharaiah [15], and Nowacki [16], to name a few. An excellent review of advances in thermo-electro-elasticity particularly those related to smart composite structures by the end of the last century can be found in Tauchert et al. [10]. Apart from the work cited in [10], a great deal of academic effort has also been made by scientists in recent years. For example, a set of general static solutions in terms of quasi-harmonic functions was developed by Chen [17], within the framework of thermo-electroelasticity. Applications of these general solutions can be found in the literatures, such as Green’s functions to crack problems [17–19], fundamental solutions for infinite/half-infinite spaces [20] and cones [21], and 3D solutions for circular/annular plates [22]. Tarn [23] developed a state-space formalism for thermo-electro-elastic analysis of a linear piezoelectric body. The mechanical behaviors of intelligent heterogeneous plates have been studied to some extent. Huang and Shen [24] examined the nonlinear and forced vibrations of FGM plates with piezoelectric layers in a thermal environment. Ootao and Tanigawa [25] presented theoretical analysis of a 3D transient thermo-electro-elastic solution for a functionally graded rectangular plate bonded to a piezoelectric plate due to partial heat supply. However, there are few works available in the literature concerning the axisymmetric solutions for FGPM circular plates subjected to external thermal loading. The present article attempts to study the thermo-electroelastic field in a transversely isotropic FGPM circular plate subjected to thermal loadings on the upper and lower


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surfaces. First, the temperature field in the plate is determined by integration. Then, the temperature induced electro-elastic field is derived by means of direct displacement method, where the displacements and electric potential are expanded as series of the radial coordinate with corresponding coefficients being functions of the axial coordinate. These functions are determined by a step-by-step integration procedure along with stress and electric displacement conditions prescribed on the upper and lower boundaries of the plate. The boundary conditions prescribed on the cylindrical edges, which are employed to determine the integral constants, are satisfied in the Saint Venant’s sense. All of the physical quantities can be generally presented explicitly for the heterogeneity models popular in the literature. The present weak form solutions can be degenerated to thermo-elastic ones by setting corresponding constants to zero. Numerical calculations are carried out to investigate the influence of material heterogeneity and the piezoelectric effect. The present solutions make sense for all the FG models where the material coefficients can vary independently in an arbitrary but continuous manner, provided that certain integral conditions and positive definite conditions are met. As a result, the solution can serve as benchmarks for various numerical simulations and theoretical approximations.

Fig. 1. Horizontal (a) and vertical (b) cross sections of an FGM circular plate and the cylindrical coordinate system (r, ␪, z).

␶r z,r + r −1 ␶r z + ␴z,z = 0, Dr,r + r −1 Dr + Dz,z = 0.

(2)

In the present study, the temperature field is assumed to be a function of z only and in a steady-state. Thus, the thermal equilibrium condition should be satisfied: qz,z = 0,

(3)

where qz is the thermal flux in the z-direction: qz = −kT,z .

2. Basic Equations By referring to a cylindrical coordinate system (r, ␪, z) with the z-axis perpendicular to the isotropy plane, the constitutive relations of a thermo-electro-elasticity medium can be described by the generalized Hooke’s law [17]. In particular, the constitutive relations for an axisymmetric problem do not depend upon the coordinate ␪ and take the following forms [20, 21]: ␴r ␴␪ ␴z ␶zr Dr Dz

= = = = = =

c11 u ,r + c12 r −1 u + c13 w,z + e31 ␾,z − ␤1 T, c12 u ,r + c11 r −1 u + c13 w,z + e31 ␾,z − ␤1 T, c13 (u ,r + r −1 u) + c33 w,z + e33 ␾,z − ␤3 T, c44 (u ,z + w,r ) + e15 ␾,r , e15 (u ,z + w,r ) − ␰11 ␾,r , e31 (u ,r + r −1 u) + e33 w,z − ␰33 ␾,z + p3 T,

where the comma represents the differentiation with respect to the indicated variable; T is the temperature variation with T = 0 corresponding to a state free of stress and electric displacement; ␴i (␶ij ) and u(w) are stresses and displacement components, respectively; ␾ and Di are, respectively, electric potential and electric displacement; cij , eij (␰ij ), and p3 are elastic, piezoelectric, and pyroelectric constants, respectively, and these are all the functions of z. The electro-mechanical coupling effect will vanish by setting eij = 0. In such a particular case, the corresponding problem will be decoupled into two sub-problems, which are respectively dielectric and pure elastic problems [19]. In cylindrical coordinates, the mechanical and electric equilibrium equations for static problems are: ␴r,r + ␶r z,z + r

−1

(␴r − ␴␪ ) = 0,

3. Problem Formulation and Temperature Field Consider a heterogeneous transversely isotropic circular plate in the cylindrical coordinate system (r, ␪, z), whose origin is coincident with the center of the plate. The plate is of radius a and thickness h, as shown in Figure 1. Furthermore, the thermal loads l and u are exerted, respectively, on the upper and lower surfaces, namely, T|z=h/2 = u , T|z=−h/2 = l ,

(1)

(4)

(5)

and, in addition, the upper and lower surfaces of the circular plate are free of stress and electric displacements. Thus, the mechanical and electric boundary conditions are: ␴z z=±h /2 = ␶r z z=±h /2 = 0, Dz z=±h /2 = 0.

(6)

The temperature field can be determined as a priori by solving the following governing equation: −[k(z)T,z ],z = 0,

(7)

which is derived by inserting Eq. (4) into Eq. (3). Integrating Eq. (7) twice from the lower limit −h/2, the temperature field is obtained: T = C1

z −h/2

k−1 (␩)d␩ + C2 ,

(8)

3D Thermo-Electro-Elastic Field in a Circular Plate

539

where C1 and C2 are integral constants. The integral constants C1 and C2 can be derived from the boundary Eq. (5), then the temperature field is given without details: z −1 −h/2 k (␩)d␩ T = (u − l ) h/2 + l . (9) −1 −h/2 k (␩)d␩ As a supplement, the temperature fields under two other typical temperature boundary conditions are also presented by a similar process, which is shown in Appendix A.

4. Displacement Method


(c33 w2,z + e33 ␾2,z )z=±h/2 (e33 w2,z − ␰33 ␾2,z )z=±h/2 [c44 (u 1,z + 2w2 ) + 2e15 ␾2 ]z=±h/2 (2c13 u 1 + c33 w0,z + e33 ␾0,z − ␤3 T)z=±h/2 (2e31 u 1 + e33 w0,z − ␰33 ␾0,z + p3 T)z=±h/2

= 0, = 0, = 0, = 0, = 0.

whose reasonableness is provided in Appendix B in detail. In this case, the generalized stress components are simplified as: ␴r = (c11 + c12 )u 1 + c13 (w0,z + r 2 w2,z ) +e31 (␾0,z + r 2 ␾2,z ) − ␤1 T, ␴␪ = (c11 + c12 )u 1 + c13 (w0,z + r 2 w2,z ) +e31 (␾0,z + r 2 ␾2,z ) − ␤1 T, ␴z = 2c13 u 1 + c33 (w0,z + r 2 w2,z ) + e33 (␾0,z + r 2 ␾2,z ) − ␤3 T, ␶zr = c44 r (u 1,z + 2w2 ) + 2r e15 ␾2 , Dr = e15 r (u 1,z + 2w2 ) − 2r ␰11 ␾2 , Dz = 2e31 u 1 + e33 (w0,z + r 2 w2,z ) −␰33 (␾0,z + r 2 ␾2,z ) + p3 T. (11) Introducing Eq. (11) into Eq. (2), we have: r {2c13 w2,z + 2e31 ␾2,z + [c44 (u 1,z + 2w2 ) + 2␾2 e15 ],z } = 0, [2c44 (u 1,z + 2w2 ) + 4e15 ␾2 + (2c13 u 1 + c33 w0,z +e33 ␾0,z − ␤3 T),z ] + r 2 (c33 w2,z + e33 ␾2,z ),z = 0, [2e15 (u 1,z + 2w2 ) − 4␰11 ␾2 + (2e31 u 1 + e33 w0,z −␰33 ␾0,z + p3 T),z ] + r 2 (e33 w2,z − ␰33 ␾2,z ),z = 0. (12) Comparing the coefficients of r m (m = 0, 1, 2) on both sides in Eq. (12), we obtain that: (c33 w2,z + e33 ␾2,z ),z = 0, (e33 w2,z − ␰33 ␾2,z ),z = 0, 2c13 w2,z + 2e31 ␾2,z + [c44 (u 1,z + 2w2 ) + 2e15 ␾2 ],z = 0, 2c44 (u 1,z + 2w2 ) + 4e15 ␾2 + (2c13 u 1 + c33 w0,z +e33 ␾0,z − ␤3 T),z = 0, 2e15 (u 1,z + 2w2 ) − 4␰11 ␾2 + (2e31 u 1 + e33 w0,z −␰33 ␾0,z + p3 T),z = 0. (13)

(15)

2 + c13 ␰33 > 0, Because of the positive definite condition e33 there exists only trivial solutions:

w2,z = 0, ␾2,z = 0. (10)

(14)

Integrating Eq. (13)1,2 once and taking advantage of Eq. (14)1,2 , we arrive at: c33 w2,z + e33 ␾2,z = 0, e33 w2,z − ␰33 ␾2,z = 0.

To obtain the thermo-electro-elastic field in the FGPM plate, the appropriate generalized displacements are assumed as: u(r, z) = r u 1 (z), w(r, z) = w0 (z) + r 2 w2 (z), ␾(r, z) = ␾0 (z) + r 2 ␾2 (z),

Substituting Eq. (11) into Eq. (6) and comparing the coefficients of r m (m = 0, 1, 2), we can get the corresponding boundary conditions:

(16)

From Eq. (13)3 , (14)3 , and (16), we have: c44 (u 1,z + 2w2 ) + 2e15 ␾2 = 0.

(17)

Integrating Eq. (13)5 once with respect to z and using the boundary condition Eq. (14)5 results in:

h/2

−h/2

[e15 (u 1,z + 2w2 ) − 2␰11 ␾2 ]dz = 0.

(18)

From Eq. (17) and (18), we can immediately derive that:

h/2 −h/2

2 e15 + ␰11 ␾2 dz = 0. c44

(19)

2 With the help of ␰11 + e15 /c44 > 0, we obtain from Eqs. (16)2 and (19) that:

␾2 = 0.

(20)

Then, the differential equations are derived from Eq. (13) with the aid of (14): w2,z u 1,z + 2w2 2c13 u 1 + c33 w0,z + e33 ␾0,z − ␤3 T 2e31 u 1 + e33 w0,z − ␰33 ␾0,z + p3 T

= 0, = 0, = 0, = 0.

(21)

From Eq. (21), the expressions of generalized displacement can be deduced as: w2 u1 w0 ␾0

= a1 , = −2a1 z + a2 , = 4a1 f1 (z) − 2a2 f0 (z) + f2 (z) + a4 , = −4a1 g1 (z) + 2a2 g0 (z) + f3 (z) + a3 ,

(22)

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P. Li et al.

where ai (i = 1 − 4) are integral constants, and

6. Integral Constants

c33 (␩)e31 (␩) − c13 (␩)e33 (␩) d␩, (i = 0, 1), 2 e33 (␩) + ␰33 (␩)c33 (␩) −h/2 z c13 (␩)␰33 (␩) + e31 (␩)e33 (␩) d␩, (i = 0, 1), ␩i fi (z) = 2 e33 (␩) + ␰33 (␩)c33 (␩) −h/2 z ␰33 (␩)␤3 (␩) − e33 (␩) p3 (␩) d␩, f2 (z) = T(␩) 2 e33 (␩) + ␰33 (␩)c33 (␩) −h/2 z e33 (␩)␤3 (␩) + c33 (␩) p3 (␩) d␩. (23) f3 (z) = T(␩) 2 e33 (␩) + ␰33 (␩)c33 (␩) −h/2

gi (z) =

z

␩i

The integral constants ai (i = 1 – 4) can be determined by the cylindrical boundary conditions at r = a.


5. Electroelastic Field From the analyses above, we can readily get the elastic and electric fields. Substitution of Eq. (22) into Eq. (10) gives: u(r, z) = r (a2 − 2a1 z), w(r, z) = a1 [r 2 + 4 f1 (z)] − 2a2 f0 (z) + f2 (z) + a4 , ␾(r, z) = −4a1 g1 (z) + 2a2 g0 (z) + f3 (z) + a3 .

6.1. Simply Supported Plate (SSP) For the simply supported plate, the cylindrical boundary conditions are [6]: N = 0, M = 0, w(a, 0) = 0, ␾(a, 0) = 0.

(24)

(29)

Substituting Eq. (27) into the first two conditions, we can derive that: −2a1 N1 + a2 N0 + N0T = 0, −2a1 N2 + a2 N1 + N1T = 0, (30) which leads to: a1 =

Then the stresses and electric displacements turn out to be: ␴r = ␴␪ = (−2a1 z + a2 )h 1 (z) + T(z)h 2 (z), ␴z = ␶zr = 0, Dr = Dz = 0,

The thermo-electro-elastic field has not been determined completely, since the integral constants ai (i = 1 − 4) are still unknown. In this section we consider two kinds of cylindrical boundary conditions at r = a.

−N1T N0 + N0T N1 −N1T N1 + N0T N2 , a2 = . 2 2(N1 − N0 N2 ) N12 − N0 N2

(31)

Making use of Eqs. (24)2 and (29)3 , we determine a4 as: a4 = −a1 [a 2 + 4 f1 (0)] + 2a2 f0 (0) − f2 (0).

(32)

Furthermore, advantage of Eqs. (24)3 and (29)4 yields: (25)

a3 = 4a1 g1 (0) − 2a2 g0 (0) − f3 (0).

(33)

where the functions h i (z), (i = 1, 2) are defined as: h 1 (z) =

2 (c11 +c12 )(e33 +c33 ␰33 )−2c13 (c13 ␰33 +e31 e33 )+2e31 (c33 e31 −c13 e33 ) , 2 e33 +␰33 c33

h 2 (z) =

2 c13 (␰33 ␤3 −e33 p3 )+e31 (e33 ␤3 +c33 p3 )−␤1 (e33 +c33 ␰33 ) . 2 e33 +␰33 c33

6.2. Clamped Plate (26)

The cylindrical boundary conditions of clamped plate are different from Eq. (29) and read [6]:

Integrating ␴r from the upper and lower limits z = ±h/2, we can derive the resultant force N and moment M as:

u(a, h/2) = 0, u(a, −h/2) = 0, w(a, 0) = 0, ␾(a, 0) = 0. (34) From Eq. (24)1 and the first two boundary conditions in Eq. (34), we get:

N≡ M≡

h/2

−h/2 h/2 −h/2

␴r d␩ = −2a1 N1 + a2 N0 + N0T , ␴r ␩d␩ = −2a1 N2 + a2 N1 + N1T ,

a1 = 0, a2 = 0. (27)

From Eqs. (24)2 and (35)3 , we can fix a4 as: a4 = − f2 (0).

with Ni = NjT

=

h/2 −h/2 h/2 −h/2

a3 = − f3 (0). T(␩)h 2 (␩)␩ d␩ ( j = 0, 1).

(36)

We can determine a3 from Eqs. (24)3 and (34)4 as:

h 1 (␩)␩i d␩, (i = 0, 1, 2), j

(35)

(28)

It is noticeable that both N and M depend upon neither r nor z.

(37)

From the analyses above, we have fixed all the integral constants ai (i = 1 − 4) in two kinds of cylindrical support conditions. Thus, the thermo-electro-elastic field has been completely determined.


541

Table 1. Material properties of Cadmium Selenide (some data from [27]) Property

␭=0

Cadmium Selenide

Elastic [109 Nm−2]

6

−1

−2

Thermal [10 N K m ] Piezoelectric [C m−2] Dielectric [10−11 C2 N−1 m−2] Pyroelectric [10−6 C N−1] Heat conduction [W K−1 m−1] Thermal expansion [10−6 K−1]


Table 2. The distribution of dimensionless temperature T along the z-axis

0 0 c11 = 74.1, c12 = 45.2, 0 0 = 83.6, c13 = 39.3, c33 0 = 13.2 c44 ␤10 = 0.621, ␤30 = 0.551 0 0 e31 = −0.16, e33 = 0.347, 0 e15 = −0.138 0 0 ␰11 = 8.26, ␰33 = 9.03 0 p3 = −2.94 k30 = 9.0 ␣r0 = 4.4

It should be pointed out that the solutions presented in aforementioned sections exactly satisfy the upper and lower boundary conditions and approximately meet the conditions on cylindrical surfaces. According to the classic monograph of Fung [26], the solutions thus obtained are of work forms and can be accepted as benchmarks for various approximate analyses.

7. Numerical Results and Discussions

z

Present

Ref. [22]

Ref. [28]

Present

Ref. [28]

−0.15 −0.12 −0.09 −0.06 −0.03 0.0 0.03 0.06 0.09 0.12 0.15

5.0000 4.5999 4.2002 3.8000 3.3999 2.9998 2.6001 2.2000 1.7998 1.4001 1.0000

5.0000 4.5999 4.2002 3.8000 3.3999 2.9998 2.6001 2.2000 1.7998 1.4001 1.0000

5.0000 4.5999 4.2002 3.8000 3.3999 2.9998 2.6001 2.2000 1.7998 1.4001 1.0000

5.0000 4.3978 3.8529 3.3599 2.9138 2.5102 2.1449 1.8144 1.5154 1.2448 1.0000

5.0000 4.3978 3.8529 3.3599 2.9138 2.5102 2.1449 1.8144 1.5154 1.2448 1.0000

stants are listed in Table 1 [27]. Consider an SS plate of height h = 0.3 [m] and radius a = 1 [m], under the action of the thermal loads u = 30[K], l = 150[K] on the upper and lower surfaces, respectively. For the sake of presentation, we introduce the following dimensionless quantities: h r z T u , r = , z = , T = , u = , a a a u a␣r0 u 0 ␰11 ␾ w ␴r , ␴ = , ␾ = . (39) w = r 0 0 a␣r0 u c33 ␣r0 u c33 a␣r0 u h =

This section is devoted to present the aforementioned solutions numerically. To this end, a special FGPM plate is considered. The material constants of the plate are specified by exponential functions of the axial coordinate z [6, 7]: z , P(z) = P0 exp ␭ 0.5 + h

␭=1

To evaluate the piezoelectric effects, we define that: (38)

where P0 denotes material constants of Cadmium Selenide, and ␭ is a parameter characterizing the degree of material heterogeneity. In particular, ␭ = 0 corresponds to the homogeneous material (Cadmium Selenide), whose material con-

up =

u u

, wp =

w w

, ␴rp =

␴r ␴r

.

(40)

u (w ) and ␴r are, respectively, the dimensionless radial (normal) displacement and the dimensionless radial stress in the case of thermo-elasticity, i.e., eij = 0.

Table 3. The dimensionless radial displacement u and the ratio u p at r = 0.25 ␭=0 z −0.15 −0.12 −0.09 −0.06 −0.03 0.0 0.03 0.06 0.09 0.12 0.15

␭=1

Present (eij = 0)

Ref. [22]

Present (eij = 0)

Ref. [29]

up

Present (eij = 0)

Present (eij = 0)

Ref. [29]

up

1.2500 1.1496 1.0491 0.9486 0.8481 0.7477 0.6472 0.5467 0.4463 0.3458 0.2453

1.2500 1.1496 1.0491 0.9486 0.8481 0.7477 0.6472 0.5467 0.4463 0.3458 0.2453

1.2940 1.1900 1.0860 0.9820 0.8780 0.7740 0.6700 0.5659 0.4619 0.3579 0.2539

1.2940 1.1900 1.0860 0.9820 0.8780 0.7740 0.6700 0.5659 0.4619 0.3579 0.2539

0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966

1.2994 1.2416 1.1837 1.1258 1.0679 1.0101 0.9522 0.8943 0.8365 0.7786 0.7207

1.3451 1.2852 1.2253 1.1654 1.1055 1.0456 0.9857 0.9258 0.8659 0.8060 0.7461

1.3451 1.2852 1.2253 1.1654 1.1055 1.0456 0.9857 0.9258 0.8659 0.8060 0.7461

0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966 0.966

542

P. Li et al.

Table 4. The dimensionless axial displacement w and the ratio w p at r = 0


␭=0

␭=1

z

Present (eij = 0)

Ref. [22]

Present (eij = 0)

Ref. [29]

wp

Present (eij = 0)

Present (eij = 0)

Ref. [29]

wp

−0.15 −0.12 −0.09 −0.06 −0.03 0.0 0.03 0.06 0.09 0.12 0.15

−6.8124 −6.7224 −6.6399 −6.5650 −6.4974 −6.4377 −6.3850 −6.3399 −6.3029 −6.2727 −6.2497

−6.8124 −6.7224 −6.6399 −6.5650 −6.4974 −6.4377 −6.3850 −6.3399 −6.3029 −6.2727 −6.2497

−6.9930 −6.9125 −6.8388 −6.7717 −6.7113 −6.6580 −6.6108 −6.5705 −6.5374 −6.5104 −6.4897

−6.9930 −6.9125 −6.8388 −6.7717 −6.7113 −6.6580 −6.6108 −6.5705 −6.5374 −6.5104 −6.4897

0.9742 0.9725 0.9709 0.9695 0.9681 0.9669 0.9658 0.9649 0.9641 0.9635 0.9630

−4.4826 −4.3829 −4.2594 −4.1110 −3.9365 −3.7347 −3.5045 −3.2444 −2.9533 −2.6299 −2.2728

−4.5631 −4.4735 −4.3598 −4.2208 −4.0555 −3.8625 −3.6407 −3.3889 −3.1056 −2.7898 −2.4400

−4.5631 −4.4735 −4.3598 −4.2208 −4.0555 −3.8625 −3.6407 −3.3889 −3.1056 −2.7898 −2.4400

0.9824 0.9797 0.9770 0.9740 0.9707 0.9669 0.9626 0.9574 0.9510 0.9427 0.9315

Tables 2–6, respectively, list the dimensionless temperature T , the radial displacement u at r = 0.25, the axial displacement w at r = 0, the radial stress ␴r , and the electric potential ␾ , for both homogeneous and heterogeneous plates. To estip mate the piezoelectric effect, the ratios u p , w p , and ␴r are presented in Tables 3, 4, and 5 as well. From Tables 2–6, we can find that the present results agree well with those in [22, 28, 29]. Furthermore, the present solutions can be reduced to thermo-elastic ones by setting eij = 0, and in this case, the preset solutions are consistent with those in [29], whose validity has been checked analytically and numerically. In short, all the physical quantities are in agreement with those in the literature, hence validating the present analyses. Figure 2 depicts the variations of dimensionless temperature T with dimensionless axial coordinate z . From Figure 2, it is seen that the temperature decreases with the gradient index ␭ at the same location. The dimensionless radial displacement u at r = 0.5 is plotted in Figure 3, which clearly shows that the radial displacement u varies linearly with z and increases with ␭. Figure 4 illustrates the axial displacement w at r = 0. It is seen that w refuses to be a constant

Table 6. The dimensionless electric potential ␾ versus z z −0.15 −0.12 −0.09 −0.06 −0.03 0.0 0.03 0.06 0.09 0.12 0.15

Present (eij = 0)

Ref. [22]

0.1589 0.1207 0.0857 0.0539 0.0253 0.0000 −0.0225 −0.0411 −0.0571 −0.0699 −0.0791

0.1589 0.1207 0.0857 0.0539 0.0253 0.0000 −0.0225 −0.0411 −0.0571 −0.0699 −0.0791

Note: Data are for homogeneous material ␭ = 0.

Table 5. The dimensionless radial stress ␴r and the ratio ␴r

p

z

Present (eij = 0)

Present (eij = 0)

Ref. [29]

␴r

−0.15 −0.12 −0.09 −0.06 −0.03 0.0 0.03 0.06 0.09 0.12 0.15

0.2046 0.1263 0.0469 −0.0292 −0.0957 −0.1439 −0.1621 −0.1349 −0.0419 0.1433 0.4548

0.2059 0.1271 0.0472 −0.0294 −0.0963 −0.1448 −0.1631 −0.1357 −0.0421 0.1442 0.4577

0.2059 0.1271 0.0472 −0.0294 −0.0963 −0.1448 −0.1631 −0.1357 −0.0421 0.1442 0.4577

0.9936 0.9936 0.9936 0.9936 0.9936 0.9936 0.9936 0.9936 0.9936 0.9936 0.9936

Note: Data are for heterogeneous material ␭ = 1.

p

Fig. 2. The dimensionless temperature T as a function of the dimensionless coordinate z .



543

Fig. 3. The variations of dimensionless radial displacement u with dimensionless axial coordinate z at r = 0.5.

Fig. 6. The distributions of dimensionless radial stress ␴r along the thickness of the plate.

Fig. 4. The dimensionless axial displacement w at r = 0, as a function of dimensionless axial coordinate z .

Fig. 7. The distribution of u along thickness of the plate.

Fig. 5. The dimensionless electric potential ␾ as a function of z .

Fig. 8. The ratio w p as a function of z = 0. Data are for r = 0.

544

P. Li et al. The weak form solutions thus obtained can be degenerated to the isotropic materials by letting [1]: c11 (z) = c33 (z) = ␭(z) + 2␮(z), c12 (z) = c13 (z) = ␭(z), c44 (z) = ␮(z), ␭1 (z) = ␭3 (z) = [3␭(z) + 2␮(z)]␣(z), (41)


Fig. 9. The ratio ␴r as a function of z = 0.

throughout the thickness as the hypothesis made in the classic plate theory. The variation of the electric potential ␾ with z is presented in Figure 5. As a posterior check of the cylindrical boundary condition, the electric potential ␾ is zero for all the heterogeneous parameter ␭s. It is also seen that ␾ > 0(␾ < 0) for a negative (positive) z and the absolute value |␾ | increases with ␭. Figure 6 depicts the distribution of the dimensionless radial stress ␴r . From Figure 6, we can see that ␴r changes significantly with ␭. In particular, the radial stress vanishes for homogeneous materials. To show the piezoelectric effect, the ratios u p , w p , and p ␴r are plotted in Figures 7–9. All of these three parameters are smaller than unit, which means that the radial and axial displacements and the radial stress are smaller than their counterparts in a pure elastic plate. Numerical results also p reveal that u p and ␴r , independent of ␭ and z, turns out to be constants, which can be verified theoretically as shown in Appendix B. On the other hand, w p are functions of both ␭ and z (see Figure 8). It is interesting that at a certain position (z ≈ 0.075), the piezoelectric effect embodied in the axial displacement are identical, regardless of the parameter ␭. In addition, w p decreases with z, i.e., the piezoelectric effect is more apparent at z = −h/2 than that at z = h/2.

8. Conclusions This article successfully solves the axisymmetric problem of a circular plate of functional graded materials with transverse isotropy, subjected to uniform thermal loads on the upper and lower surfaces. The temperature field is determined as a priori through an integration procedure. The electro-elastic field induced by the temperature field is presented by a direct displacement method. The boundary conditions on the upper and lower surfaces of the circular plate are satisfied exactly and the cylindrical boundary conditions at the edge of the plate are approximately met in the Saint Venant’s sense.

where ␣(z) is the coefficient of thermal expansion, and ␭(z) and ␮(z) are the Lam´e elastic constants. A numeral example is given for a particular FGPM model. The present solutions are validated comparing those available in the literature. The effect of material heterogeneity and the piezoelectric effect have been shown. It should be pointed out that the present solution is valid for any FGPM model in which each material constant can vary independently in a continuous way, provided that the positive definite conditions have been satisfied.

References [1] H.J. Ding and W.Q. Chen, Three Dimensional Problems of Piezoelasticity, Nova Science Publisher, New York, 2000. [2] M. Koizumi, Concept of FGM, Ceram. Trans., vol. 34, pp. 3–10, 1993. [3] X.H. Zhu and Z.Y. Meng, Operational principle, fabrication and displacement characteristic of a functionally gradient piezoelectric ceramic actuator, Sens. Actuator A, vol. 48, pp. 169–176, 1995. [4] J.N. Reddy, C.M. Wang, and S. Kitipornchai, Axisymmetric bending of functionally graded circular and annular plates, Eur. J. Mech. A. Solids, vol. 18, pp. 185–199, 1999. [5] J.N. Reddy and Z.G. Cheng, Three-dimensional thermomechanical deformations of functionally graded rectangular plates, Eur. J. Mech. A. Solids, vol. 20, pp. 841–855, 2001. [6] Y. Wang, R.Q. Xu, and H.J. Ding, Analytical solutions of functionally graded piezoelectric circular plates subjected to axisymmetric loads, Acta Mech., vol. 215, pp. 387–305, 2010. [7] X.Y. Li, J. Wu, H.J. Ding, and W.Q. Chen, 3D analytical solution for a functionally graded transversely isotropic piezoelectric circular plate under tension and bending, Int. J. Eng. Sci., vol. 49, pp. 664–676, 2011. [8] E.M. Sekouri, Y.R. Hu, and A.D. Ngo, Modeling of a circular plate with piezoelectric actuators, Mechatronics, vol. 14, pp. 1007–1020, 2004. [9] C.H.J. Fox, X. Chen, and S. McWilliam, Analysis of the deflection of a circular plate with an annular piezoelectric actuator, Sens. Actuators, A, vol. 133, pp. 180–194, 2007. [10] T.R. Tauchert, F. Ashida, N. Noda, S. Adali, and V. Verijenko, Developments in thermo-piezo-elasticity with relevance to smart composite structures, Compos. Struct., vol. 48, pp. 31–38, 2000. [11] F. Ashida and T.R. Tauchert, Transient response of a piezothermoelastic circular disk under axisymmetric heating, Acta Mech., vol. 128, pp. 1–14, 1998. [12] R.D. Mindlin, On the equations of motion of piezoelectric crystals. In: Problems of Continuum Mechanics (N. I. Muskhelishvili 70th Birthday Volume), The Society for Industrial and Applied Mathematics, Philadelphia, pp. 282–290, 1961. [13] R.D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates, Int. J. Solids Struct., vol. 10, pp. 625–637, 1974. [14] H.F. Tiersten, On the nonlinear equations of thermoelectroelasticity, Int. J. Eng. Sci., vol. 9, pp. 587–604, 1971. [15] D.S. Chandrasekharaiah, A generalized linear thermoelasticity theory for piezoelectric media, Acta Mech., vol. 71, pp. 39–49, 1988.


3D Thermo-Electro-Elastic Field in a Circular Plate [16] W. Nowacki, Some general theorems of thermopiezoelectricity, J. Therm. Stresses, vol. 1, pp. 171–182, 1978. [17] W.Q. Chen, On the general solution for piezothermoelasticity for transverse isotropy with application, ASME J. Appl. Mech., vol. 67, pp. 705–711, 2000. [18] W.Q. Chen, C.M. Lim, and H.J. Ding, Point temperature solution for a penny-shaped crack in an infinite transversely isotropic thermo-piezo-elastic medium, Eng. Anal. Boundary Elem., vol. 29, pp. 524–532, 2005. [19] X.Y. Li, On the half-infinite crack problem in thermoelectro-elasticity, Mech. Res. Commun., vol. 38, pp. 506–511, 2011. [20] P.F. Hou, W. Luo, and Y.T.A. Leung, A point heat source on the surface of a semi-infinite transversely isotropic piezothermoelastic material, J. Appl. Mech., vol. 75, pp. 110131–110138, 2008. [21] P.F. Hou, W. Luo, and Y.T.A. Leung, A point heat source on the apex of a transversely isotropic piezothermoelastic cone, Eur. J. Mech. A. Solids, vol. 27, pp. 418–428, 2008. [22] P.F. Hou, L.J. Guo, and W. Luo, A simply supported circular piezothermoelastic plate under uniform loading, Proceedings of the 2006 Symposium on Piezoelectricity, Acoustic Waves and Device Applications, pp. 40–49, December 14–16, Hangzhou, China, 2007. [23] J.Q. Tarn, A state space formalism for piezothermoelasticity, Int. J. Solids Struct., vol. 39, pp. 5173–5184, 2002. [24] X.L. Huang and H.S. Shen, Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments, J. Sound Vib., vol. 289, pp. 25–53, 2006. [25] Y. Ootao and Y. Tanigawa, Three-dimensional transient piezothermoelasticity in functionally graded rectangular plate bonded to a piezoelectric plate, Int. J. Solids Struct., vol. 37, pp. 4377–4401, 2000. [26] Y.C. Fung, Foundation of Solid Mechanics, Prentice-Hall, Englewood Cliffs, 1965. [27] H.J. Ding, F.L. Guo, and P.F. Hou, A general solution for piezothermoelasticity of transversely isotropic piezoelectric materials and its applications, Int. J. Eng. Sci., vol. 38, pp. 1415–1440, 2000. [28] L.S. Ma and T.J. Wang, Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings, Int. J. Solids Struct., vol. 40, pp. 3311–3330, 2003. [29] X.Y. Li, P.D. Li, G.Z. Kang, and D.Z. Pan, Three-dimensional thermo-elasticity field in a heterogeneous circular plate of transversely isotropic material, Math. Mech. Solids, vol. 8, pp. 464–475, 2012. [30] P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I, McGraw-Hill, New York, 1953. [31] R. Courant and D. Hilbert, Methods of Mathematical Physics (Vols. I and II), John Wiley & Sons, Inc., New York, 1989.

545 where ␤ and l are prescribed constants. Similarly, the expression of dimensionless temperature can be determined without details: l h −␭(0.5+z/ h) e − e−␭ ; 0 ␭k3

(u − l )h T(␭, z) = 1 − e−␭(z/ h+0.5) 0 −␭ h(1 − e ) − ␭k3 ␤ hl (1 − e−␭ ) − ␭␤k30 u + . h(1 − e−␭ ) − ␭k30 ␤

T(␭, z) = u −

(A.3)

(A.4)

The temperature fields in Eqs. (A.3) and (A.4) correspond to the boundary conditions (A.1) and (A.2), respectively. The coupled electro-elastic fields induced by the thermal loading in (A.1) and (A.2) can be also expressed by the previous formulae in Sections 4–6, where the temperature should be replaced by those in Eqs. (A.3) and (A.4), respectively.

Appendix B The generalized displacements (elastic displacement and electric potential) can be expanded into the power series of the radial coordinate r [31]: u(r, z) = ␾(r, z) =

∞ m=0 ∞

r m u m (z), w(r, z) =

∞

r m wm (z),

m=0

r m ␾m (z),

(B.1)

m=0

where u m (z), wm (z), and ␾m (z) are referred to as displacement functions. Substituting Eq. (B.1) in Eq. (1), we can obtain the generalized stress components as: ∞

␴r = c12

u0 r m {[(m + 1)c11 + c12 ]u m+1 − ␤1 T+ r m=0

+ c13 wm,z + e31 ␾m,z }, ∞ u0 ␴␪ = c11 − ␤1 T+ r m {[(m + 1)c12 + c11 ]u m+1 r

Appendix A

m=0

This section presents the temperature fields in the FGPM plate corresponding to Neumann boundary conditions and Newton’s law of cooling conditions exerted on a surface of the plate. The Neumann boundary conditions read [30]: qz |z=−h/2 = l , T|z=h/2 = u ,

(A.1)

where l is a constant. The latter boundary conditions are [30]: [T − ␤qz ]|z=−h/2 = l , T|z=h/2 = u ,

(A.2)

+ c13 wm,z + e31 ␾m,z }, ∞ u0 r m [(m + 2)c13 u m+1 ␴z = c13 − ␤3 T + r m=0

+ c33 wm,z + e33 ␾m,z ], ∞ ␶r z = r m [c44 u m,z + (m + 1)(c44 wm+1 + e15 ␾m+1 )], Dr =

m=0 ∞

r m [e15 u m,z + (m + 1)(e15 wm+1 − ␰11 ␾m+1 )],

m=0 ∞

Dz = e31

u0 + p3 T + r m [(m + 2)e31 u m+1 r m=0

546

P. Li et al. + e33 wm,z − ␰33 ␾m,z ].

(B.2)

No singularity of stresses at r = 0 requires u 0 = 0. As a result, the relation ␴r (0, z) = ␴␪ (0, z) = (c11 + c12 )u 1 + c13 w0,z + e31 ␾0,z − ␤1 T is established. This should be satisfied because there is no difference for stresses between the radial and circumferential directions at r = 0. Substitution of Eq. (B.2) into Eq. (2) gives rise to: ∞ m=0 ∞

u m+3 = 0.

(B.12)

Thus, we can conclude from Eqs. (B.9), (B.11), and (B.12) that: u 2m+2 = 0, w2m+1 = 0, ␾2m+1 = 0, (m = 0, 1, 2 · · ·).(B.13) This can facilitate our further analyses. In view of these facts, we can further make the following assumptions, without losing any generality, that:

mr m−1 [(m + 2)c11 u m+1 + c13 wm,z + e31 ␾m,z ]

+

From Eqs. (B.6), (B.9)1 , and (B.11), we have:

r m [c44 u m,z + (m + 1)(c44 wm+1 + e15 ␾m+1 )],z = 0,

u 3 (z) = 0, w4 (z) = 0, ␾4 (z) = 0.

(B.14)

m=0

(B.3) −(␤3 T),z +

∞

As a consequence, we can derive that:

r m [(m + 2)c13 u m+1 + c33 wm,z + e33 ␾m,z ],z

u 2m−1 = 0, w2m = 0, ␾2m = 0, (m = 2, 3, 4 · · ·). (B.15)


m=0

+

∞

(m + 1)r m−1 [c44 u m,z + (m + 1)

m=0

× (c44 wm+1 + e15 ␾m+1 )] = 0, (B.4) ∞ r m [(m + 2)e31 u m+1 + e33 wm,z − ␰33 ␾m,z ],z ( p3 T),z +

The assumptions in Eq. (B.15) are reasonable since these satisfy all the equilibrium equations as well as the boundary conditions. Hence, the appropriate generalized displacements turn out to be: u(r, z) = r u 1 (z), w(r, z) = w0 (z) + r 2 w2 (z), ␾(r, z) = ␾0 (z) + r 2 ␾2 (z).

m=0

+

∞

(m + 1)r m−1 [e15 u m,z + (m + 1)

(B.16)

m=0

× (e15 wm+1 − ␰11 ␾m+1 )] = 0.

(B.5)

Comparing the coefficients of r m on both sides in Eqs. (B.3), (B.4), and (B.5), we can easily get that: (m + 1)[(m + 3)c11 u m+2 + c13 wm+1,z + e31 ␾m+1,z ] +[c44 u m,z + (m + 1)(c44 wm+1 + e15 ␾m+1 )],z = 0, (m = 0, 1, 2 · · ·), (B.6) (m + 2)[c44 u m+1,z + (m + 2)(c44 wm+2 + e15 ␾m+2 )] +[(m + 2)c13 u m+1 + c33 wm,z + e33 ␾m,z ],z = 0, (m = 1, 2, 3 · · ·), (B.7) (m + 2)[e15 u m+1,z + (m + 2)(e15 wm+2 − ␰11 ␾m+2 )] +[(m + 2)e31 u m+1 + e33 wm,z − ␰33 ␾m,z ],z = 0, (m = 1, 2, 3 · · ·). (B.8)

Appendix C p

In this section, the ratios u p and ␴r are proven to be constants, as shown in Figures 7 and 9. From Eqs. (24)1 , (25)1 , and (31), u and ␴r can be rewritten as:

r N0T N2 − N1T N1 − z(N0T N1 − N1T N0 ) , (C.1) u= N12 − N0 N2

h 1 (z) N0T N2 − N1T N1 − z(N0T N1 − N1T N0 ) ␴r = + T(z)h 2 (z). N12 − N0 N2 (C.2) With the aid of Eq. (38), Eq. (26) takes the form as: z h 1 (z) = exp ␭ 0.5 + , h z , h 2 (z) = h 02 exp 2␭ 0.5 + h

If we assume that: u m+1 = 0, wm = 0, ␾m = 0, (m = 1, 2 · · ·),

h 01

(B.9)

we can obtain from Eqs. (B.7) and (B.8) that: c44 wm+2 + e15 ␾m+2 = 0, e15 wm+2 − ␰11 ␾m+2 = 0. (B.10) The positive definite condition

2 e15

+ c44 ␰11 > 0 leads to:

wm+2 = 0, ␾m+2 = 0.

(B.11)

(C.3)

where

h 01

=

0 0 c11 +c12

h 02 =

, (C.4)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 e33 e33 +c33 ␰33 −2c13 c13 ␰33 +e31 e33 +2e31 c33 e31 −c13 e33

0 0 0 0 e33 e33 +␰33 c33

0 0 0 0 0 0 0 0 0 0 0 0 2 c13 ␰33 ␤3 −e33 p3 +e31 e33 ␤3 +c33 p3 −␤10 e33 +c33 ␰33 0 0 0 0 e33 e33 +␰33 c33

.


(C.5)

2 h/2 h/2 2 ␩I1 (␩)d␩ − ␩ I1 (␩)d␩ I1 (␩)d␩, J3 = −h/2 −h/2 −h/2 z z , I2 = exp 2␭ 0.5 + . (C.7) I1 = exp ␭ 0.5 + h h

(C.6)

When eij = 0, Eq. (C.4) can be reduced to:

From Eqs. (C.3), (C.4), and (28), Eqs. (C.1) and (C.2) can be expressed as: r h 02 (J1 − zJ2 ) , h 01 J3 h 0 [I1 (J1 − zJ2 ) + I2 T J3 ] ␴r = 2 , J3 u=

h/2

0 0 h 01 = c11 + c12 −

where J1 =

␩ I1 (␩)d␩

−h/2 h/2

J2 =

h/2

−h/2 h/2

␩I1 (␩)d␩

−h/2 h/2

−

−h/2

T(␩)I2 (␩)d␩

−h/2 h/2

␩I1 (␩)d␩

I1 (␩)d␩

0 0 0 0 2c13 c13 c13 ␤3 0 , h = − ␤10 . 2 0 0 c33 c33

(C.8)

h/2

2

−


547

−h/2 h/2

␩T(␩)I2 (␩)d␩,

T(␩)I2 (␩)d␩

−h/2 h/2

−h/2

␩T(␩)I2 (␩)d␩,

Then, the ratios: up =

u u

=

h 02 h 01 h 01 h 02

, ␴rp =

␴r ␴r

=

h 02 h 02

.

(C.9)

Since h 01 , h 02 , h 01 , and h 02 are not related to r , z , and ␭, the p ratios u p and ␴r have been proven to be a constant.